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Theorem frgpuplem 15409
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b  |-  B  =  ( Base `  H
)
frgpup.n  |-  N  =  ( inv g `  H )
frgpup.t  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
frgpup.h  |-  ( ph  ->  H  e.  Grp )
frgpup.i  |-  ( ph  ->  I  e.  V )
frgpup.a  |-  ( ph  ->  F : I --> B )
frgpup.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpup.r  |-  .~  =  ( ~FG  `  I )
Assertion
Ref Expression
frgpuplem  |-  ( (
ph  /\  A  .~  C )  ->  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) )
Distinct variable groups:    y, z, A    y, F, z    y, N, z    y, B, z    ph, y, z    y, I, z
Allowed substitution hints:    C( y, z)    .~ ( y, z)    T( y, z)    H( y, z)    V( y, z)    W( y, z)

Proof of Theorem frgpuplem
Dummy variables  a 
b  u  v  n  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup.w . . . . . . 7  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpup.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
31, 2efgval 15354 . . . . . 6  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }
4 coeq2 5034 . . . . . . . . . . . . 13  |-  ( u  =  v  ->  ( T  o.  u )  =  ( T  o.  v ) )
54oveq2d 6100 . . . . . . . . . . . 12  |-  ( u  =  v  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) )
6 eqid 2438 . . . . . . . . . . . 12  |-  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  =  { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }
75, 6eqer 6941 . . . . . . . . . . 11  |-  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  Er  _V
87a1i 11 . . . . . . . . . 10  |-  ( ph  ->  { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  Er  _V )
9 ssv 3370 . . . . . . . . . . 11  |-  W  C_  _V
109a1i 11 . . . . . . . . . 10  |-  ( ph  ->  W  C_  _V )
118, 10erinxp 6981 . . . . . . . . 9  |-  ( ph  ->  ( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W
)
12 df-xp 4887 . . . . . . . . . . . . 13  |-  ( W  X.  W )  =  { <. u ,  v
>.  |  ( u  e.  W  /\  v  e.  W ) }
1312ineq1i 3540 . . . . . . . . . . . 12  |-  ( ( W  X.  W )  i^i  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) } )  =  ( { <. u ,  v >.  |  ( u  e.  W  /\  v  e.  W ) }  i^i  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) } )
14 incom 3535 . . . . . . . . . . . 12  |-  ( ( W  X.  W )  i^i  { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) } )  =  ( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )
15 inopab 5008 . . . . . . . . . . . 12  |-  ( {
<. u ,  v >.  |  ( u  e.  W  /\  v  e.  W ) }  i^i  {
<. u ,  v >.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) } )  =  { <. u ,  v >.  |  ( ( u  e.  W  /\  v  e.  W )  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
1613, 14, 153eqtr3i 2466 . . . . . . . . . . 11  |-  ( {
<. u ,  v >.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) }  i^i  ( W  X.  W ) )  =  { <. u ,  v >.  |  ( ( u  e.  W  /\  v  e.  W
)  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) ) }
17 vex 2961 . . . . . . . . . . . . . 14  |-  u  e. 
_V
18 vex 2961 . . . . . . . . . . . . . 14  |-  v  e. 
_V
1917, 18prss 3954 . . . . . . . . . . . . 13  |-  ( ( u  e.  W  /\  v  e.  W )  <->  { u ,  v } 
C_  W )
2019anbi1i 678 . . . . . . . . . . . 12  |-  ( ( ( u  e.  W  /\  v  e.  W
)  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) )  <->  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) )
2120opabbii 4275 . . . . . . . . . . 11  |-  { <. u ,  v >.  |  ( ( u  e.  W  /\  v  e.  W
)  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) ) }  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
2216, 21eqtri 2458 . . . . . . . . . 10  |-  ( {
<. u ,  v >.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) }  i^i  ( W  X.  W ) )  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
23 ereq1 6915 . . . . . . . . . 10  |-  ( ( { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  ->  (
( { <. u ,  v >.  |  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W  <->  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W
) )
2422, 23ax-mp 5 . . . . . . . . 9  |-  ( ( { <. u ,  v
>.  |  ( H  gsumg  ( T  o.  u ) )  =  ( H 
gsumg  ( T  o.  v
) ) }  i^i  ( W  X.  W
) )  Er  W  <->  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W
)
2511, 24sylib 190 . . . . . . . 8  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  Er  W )
26 simplrl 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x  e.  W )
27 fviss 5787 . . . . . . . . . . . . . . . 16  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
281, 27eqsstri 3380 . . . . . . . . . . . . . . 15  |-  W  C_ Word  ( I  X.  2o )
2928, 26sseldi 3348 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x  e. Word  ( I  X.  2o ) )
30 opelxpi 4913 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  b >.  e.  ( I  X.  2o ) )
3130adantl 454 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <. a ,  b >.  e.  ( I  X.  2o ) )
32 simprl 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
a  e.  I )
33 2oconcl 6750 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
3433ad2antll 711 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( 1o  \  b
)  e.  2o )
35 opelxpi 4913 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  I  /\  ( 1o  \  b
)  e.  2o )  ->  <. a ,  ( 1o  \  b )
>.  e.  ( I  X.  2o ) )
3632, 34, 35syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <. a ,  ( 1o 
\  b ) >.  e.  ( I  X.  2o ) )
3731, 36s2cld 11838 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  e. Word  (
I  X.  2o ) )
38 splcl 11786 . . . . . . . . . . . . . 14  |-  ( ( x  e. Word  ( I  X.  2o )  /\  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  e. Word  (
I  X.  2o ) )  ->  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  e. Word  ( I  X.  2o ) )
3929, 37, 38syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e. Word  ( I  X.  2o ) )
401efgrcl 15352 . . . . . . . . . . . . . . 15  |-  ( x  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4126, 40syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
4241simprd 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  W  = Word  ( I  X.  2o ) )
4339, 42eleqtrrd 2515 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W )
4426, 43jca 520 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x  e.  W  /\  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) )
45 swrdcl 11771 . . . . . . . . . . . . . . . . . 18  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. 0 ,  n >. )  e. Word  ( I  X.  2o ) )
4629, 45syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o ) )
47 frgpup.b . . . . . . . . . . . . . . . . . . 19  |-  B  =  ( Base `  H
)
48 frgpup.n . . . . . . . . . . . . . . . . . . 19  |-  N  =  ( inv g `  H )
49 frgpup.t . . . . . . . . . . . . . . . . . . 19  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
50 frgpup.h . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  H  e.  Grp )
51 frgpup.i . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  I  e.  V )
52 frgpup.a . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F : I --> B )
5347, 48, 49, 50, 51, 52frgpuptf 15407 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
5453ad2antrr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  T : ( I  X.  2o ) --> B )
55 ccatco 11809 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( (
x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  =  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )
5646, 37, 54, 55syl3anc 1185 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  =  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )
5756oveq2d 6100 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
5850ad2antrr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  H  e.  Grp )
59 grpmnd 14822 . . . . . . . . . . . . . . . . 17  |-  ( H  e.  Grp  ->  H  e.  Mnd )
6058, 59syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  H  e.  Mnd )
61 wrdco 11805 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( x substr  <.
0 ,  n >. ) )  e. Word  B )
6246, 54, 61syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B
)
63 wrdco 11805 . . . . . . . . . . . . . . . . 17  |-  ( (
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. ">  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  B )
6437, 54, 63syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  e. Word  B
)
65 eqid 2438 . . . . . . . . . . . . . . . . 17  |-  ( +g  `  H )  =  ( +g  `  H )
6647, 65gsumccat 14792 . . . . . . . . . . . . . . . 16  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B  /\  ( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  e. Word  B
)  ->  ( H  gsumg  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
6760, 62, 64, 66syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
6854, 31, 36s2co 11872 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  =  <" ( T `  <. a ,  b >. )
( T `  <. a ,  ( 1o  \ 
b ) >. ) "> )
69 df-ov 6087 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7069a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T b )  =  ( T `
 <. a ,  b
>. ) )
71 df-ov 6087 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) b )  =  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )
72 eqid 2438 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
7372efgmval 15349 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )
b )  =  <. a ,  ( 1o  \ 
b ) >. )
7471, 73syl5eqr 2484 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )  =  <. a ,  ( 1o  \  b )
>. )
7574adantl 454 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )  =  <. a ,  ( 1o  \  b )
>. )
7675fveq2d 5735 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) `  <. a ,  b >. ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
)
7747, 48, 49, 50, 51, 52, 72frgpuptinv 15408 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  <. a ,  b >.  e.  (
I  X.  2o ) )  ->  ( T `  ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. ) `  <. a ,  b >. )
)  =  ( N `
 ( T `  <. a ,  b >.
) ) )
7830, 77sylan2 462 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) `  <. a ,  b >. ) )  =  ( N `  ( T `  <. a ,  b >. ) ) )
7978adantlr 697 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
) `  <. a ,  b >. ) )  =  ( N `  ( T `  <. a ,  b >. ) ) )
8076, 79eqtr3d 2472 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  <. a ,  ( 1o  \ 
b ) >. )  =  ( N `  ( T `  <. a ,  b >. )
) )
8169fveq2i 5734 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N `
 ( a T b ) )  =  ( N `  ( T `  <. a ,  b >. ) )
8280, 81syl6reqr 2489 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( N `  (
a T b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
8370, 82s2eqd 11831 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  <" ( a T b ) ( N `
 ( a T b ) ) ">  =  <" ( T `  <. a ,  b >. ) ( T `
 <. a ,  ( 1o  \  b )
>. ) "> )
8468, 83eqtr4d 2473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> )  =  <" ( a T b ) ( N `  ( a T b ) ) "> )
8584oveq2d 6100 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> ) )  =  ( H  gsumg 
<" ( a T b ) ( N `
 ( a T b ) ) "> ) )
86 simprr 735 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
b  e.  2o )
8754, 32, 86fovrnd 6221 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T b )  e.  B )
8847, 48grpinvcl 14855 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  e.  Grp  /\  ( a T b )  e.  B )  ->  ( N `  ( a T b ) )  e.  B
)
8958, 87, 88syl2anc 644 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( N `  (
a T b ) )  e.  B )
9047, 65gsumws2 14793 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H  e.  Mnd  /\  ( a T b )  e.  B  /\  ( N `  ( a T b ) )  e.  B )  -> 
( H  gsumg 
<" ( a T b ) ( N `
 ( a T b ) ) "> )  =  ( ( a T b ) ( +g  `  H
) ( N `  ( a T b ) ) ) )
9160, 87, 89, 90syl3anc 1185 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg 
<" ( a T b ) ( N `
 ( a T b ) ) "> )  =  ( ( a T b ) ( +g  `  H
) ( N `  ( a T b ) ) ) )
92 eqid 2438 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0g
`  H )  =  ( 0g `  H
)
9347, 65, 92, 48grprinv 14857 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H  e.  Grp  /\  ( a T b )  e.  B )  ->  ( ( a T b ) ( +g  `  H ) ( N `  (
a T b ) ) )  =  ( 0g `  H ) )
9458, 87, 93syl2anc 644 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( a T b ) ( +g  `  H ) ( N `
 ( a T b ) ) )  =  ( 0g `  H ) )
9585, 91, 943eqtrd 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> ) )  =  ( 0g `  H
) )
9695oveq2d 6100 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( 0g `  H ) ) )
9747gsumwcl 14791 . . . . . . . . . . . . . . . . . 18  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B
)  ->  ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) )  e.  B
)
9860, 62, 97syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) )  e.  B )
9947, 65, 92grprid 14841 . . . . . . . . . . . . . . . . 17  |-  ( ( H  e.  Grp  /\  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) )  e.  B )  ->  (
( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) ( +g  `  H ) ( 0g `  H
) )  =  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) )
10058, 98, 99syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( 0g `  H ) )  =  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) )
10196, 100eqtrd 2470 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) )  =  ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) )
10257, 67, 1013eqtrrd 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) )  =  ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) )
103102oveq1d 6099 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
104 swrdcl 11771 . . . . . . . . . . . . . . . 16  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. n ,  ( # `  x ) >. )  e. Word  ( I  X.  2o ) )
10529, 104syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o ) )
106 wrdco 11805 . . . . . . . . . . . . . . 15  |-  ( ( ( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
)  e. Word  B )
107105, 54, 106syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)
10847, 65gsumccat 14792 . . . . . . . . . . . . . 14  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
x substr  <. 0 ,  n >. ) )  e. Word  B  /\  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)  ->  ( H  gsumg  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
x substr  <. 0 ,  n >. ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
10960, 62, 107, 108syl3anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) )  =  ( ( H  gsumg  ( T  o.  ( x substr  <. 0 ,  n >. ) ) ) ( +g  `  H
) ( H  gsumg  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) ) )
110 ccatcl 11748 . . . . . . . . . . . . . . . 16  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  e. Word  ( I  X.  2o ) )  ->  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o ) )
11146, 37, 110syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o ) )
112 wrdco 11805 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  e. Word  B
)
113111, 54, 112syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  e. Word  B
)
11447, 65gsumccat 14792 . . . . . . . . . . . . . 14  |-  ( ( H  e.  Mnd  /\  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) )  e. Word  B  /\  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) )  e. Word  B
)  ->  ( H  gsumg  ( ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
11560, 113, 107, 114syl3anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )  =  ( ( H  gsumg  ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) ) ( +g  `  H ) ( H  gsumg  ( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) ) )
116103, 109, 1153eqtr4d 2480 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) )  =  ( H  gsumg  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) ) )
117 simplrr 739 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  n  e.  ( 0 ... ( # `  x
) ) )
118 elfzuz 11060 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 0 ... ( # `  x
) )  ->  n  e.  ( ZZ>= `  0 )
)
119 eluzfz1 11069 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... n
) )
120117, 118, 1193syl 19 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
0  e.  ( 0 ... n ) )
121 lencl 11740 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e. Word  ( I  X.  2o )  ->  ( # `  x )  e.  NN0 )
12229, 121syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  NN0 )
123 nn0uz 10525 . . . . . . . . . . . . . . . . . . 19  |-  NN0  =  ( ZZ>= `  0 )
124122, 123syl6eleq 2528 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  ( ZZ>= `  0
) )
125 eluzfz2 11070 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  x )  e.  ( ZZ>= `  0 )  ->  ( # `  x
)  e.  ( 0 ... ( # `  x
) ) )
126124, 125syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( # `  x )  e.  ( 0 ... ( # `  x
) ) )
127 ccatswrd 11778 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e. Word  ( I  X.  2o )  /\  ( 0  e.  ( 0 ... n )  /\  n  e.  ( 0 ... ( # `  x ) )  /\  ( # `  x )  e.  ( 0 ... ( # `  x
) ) ) )  ->  ( ( x substr  <. 0 ,  n >. ) concat 
( x substr  <. n ,  ( # `  x
) >. ) )  =  ( x substr  <. 0 ,  ( # `  x
) >. ) )
12829, 120, 117, 126, 127syl13anc 1187 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
)  =  ( x substr  <. 0 ,  ( # `  x ) >. )
)
129 swrdid 11777 . . . . . . . . . . . . . . . . 17  |-  ( x  e. Word  ( I  X.  2o )  ->  ( x substr  <. 0 ,  ( # `  x ) >. )  =  x )
13029, 129syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x substr  <. 0 ,  ( # `  x
) >. )  =  x )
131128, 130eqtrd 2470 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( ( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
)  =  x )
132131coeq2d 5038 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
) )  =  ( T  o.  x ) )
133 ccatco 11809 . . . . . . . . . . . . . . 15  |-  ( ( ( x substr  <. 0 ,  n >. )  e. Word  (
I  X.  2o )  /\  ( x substr  <. n ,  ( # `  x
) >. )  e. Word  (
I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  ( (
x substr  <. 0 ,  n >. ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )  =  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) )
13446, 105, 54, 133syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( x substr  <. 0 ,  n >. ) concat  ( x substr  <.
n ,  ( # `  x ) >. )
) )  =  ( ( T  o.  (
x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
135132, 134eqtr3d 2472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  x
)  =  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
136135oveq2d 6100 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  x
) )  =  ( H  gsumg  ( ( T  o.  ( x substr  <. 0 ,  n >. ) ) concat  ( T  o.  ( x substr  <.
n ,  ( # `  x ) >. )
) ) ) )
137 splval 11785 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  W  /\  ( n  e.  (
0 ... ( # `  x
) )  /\  n  e.  ( 0 ... ( # `
 x ) )  /\  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  e. Word  ( I  X.  2o ) ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )
13826, 117, 117, 37, 137syl13anc 1187 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )
139138coeq2d 5038 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  =  ( T  o.  ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
140 ccatco 11809 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> )  e. Word  ( I  X.  2o )  /\  (
x substr  <. n ,  (
# `  x ) >. )  e. Word  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  -> 
( T  o.  (
( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )  =  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> ) ) concat 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) )
141111, 105, 54, 140syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) concat  ( x substr  <. n ,  ( # `  x
) >. ) ) )  =  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> ) ) concat 
( T  o.  (
x substr  <. n ,  (
# `  x ) >. ) ) ) )
142139, 141eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  =  ( ( T  o.  (
( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) )
143142oveq2d 6100 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )  =  ( H  gsumg  ( ( T  o.  ( ( x substr  <. 0 ,  n >. ) concat  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> ) ) concat  ( T  o.  ( x substr  <. n ,  ( # `  x
) >. ) ) ) ) )
144116, 136, 1433eqtr4d 2480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( H  gsumg  ( T  o.  x
) )  =  ( H  gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
145 vex 2961 . . . . . . . . . . . 12  |-  x  e. 
_V
146 ovex 6109 . . . . . . . . . . . 12  |-  ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  e.  _V
147 eleq1 2498 . . . . . . . . . . . . . . 15  |-  ( u  =  x  ->  (
u  e.  W  <->  x  e.  W ) )
148 eleq1 2498 . . . . . . . . . . . . . . 15  |-  ( v  =  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  ( v  e.  W  <->  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) )
149147, 148bi2anan9 845 . . . . . . . . . . . . . 14  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( (
u  e.  W  /\  v  e.  W )  <->  ( x  e.  W  /\  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) ) )
15019, 149syl5bbr 252 . . . . . . . . . . . . 13  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( {
u ,  v } 
C_  W  <->  ( x  e.  W  /\  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W ) ) )
151 coeq2 5034 . . . . . . . . . . . . . . 15  |-  ( u  =  x  ->  ( T  o.  u )  =  ( T  o.  x ) )
152151oveq2d 6100 . . . . . . . . . . . . . 14  |-  ( u  =  x  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  x
) ) )
153 coeq2 5034 . . . . . . . . . . . . . . 15  |-  ( v  =  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  ( T  o.  v )  =  ( T  o.  ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) )
154153oveq2d 6100 . . . . . . . . . . . . . 14  |-  ( v  =  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  ( H  gsumg  ( T  o.  v ) )  =  ( H 
gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
155152, 154eqeqan12d 2453 . . . . . . . . . . . . 13  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) )  <->  ( H  gsumg  ( T  o.  x ) )  =  ( H 
gsumg  ( T  o.  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) ) )
156150, 155anbi12d 693 . . . . . . . . . . . 12  |-  ( ( u  =  x  /\  v  =  ( x splice  <.
n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  ( ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) )  <->  ( (
x  e.  W  /\  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W )  /\  ( H  gsumg  ( T  o.  x ) )  =  ( H  gsumg  ( T  o.  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) ) ) )
157 eqid 2438 . . . . . . . . . . . 12  |-  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }
158145, 146, 156, 157braba 4475 . . . . . . . . . . 11  |-  ( x { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  ( ( x  e.  W  /\  (
x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  e.  W )  /\  ( H  gsumg  ( T  o.  x ) )  =  ( H  gsumg  ( T  o.  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) ) )
15944, 144, 158sylanbrc 647 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  W  /\  n  e.  ( 0 ... ( # `  x
) ) ) )  /\  ( a  e.  I  /\  b  e.  2o ) )  ->  x { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
160159ralrimivva 2800 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  W  /\  n  e.  ( 0 ... ( # `
 x ) ) ) )  ->  A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
161160ralrimivva 2800 . . . . . . . 8  |-  ( ph  ->  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )
162 fvex 5745 . . . . . . . . . . 11  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
1631, 162eqeltri 2508 . . . . . . . . . 10  |-  W  e. 
_V
164 erex 6932 . . . . . . . . . 10  |-  ( {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  ->  ( W  e.  _V  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  _V ) )
16525, 163, 164ee10 1386 . . . . . . . . 9  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  _V )
166 ereq1 6915 . . . . . . . . . . 11  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( r  Er  W  <->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  Er  W ) )
167 breq 4217 . . . . . . . . . . . . 13  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
1681672ralbidv 2749 . . . . . . . . . . . 12  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  A. a  e.  I  A. b  e.  2o  x { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
1691682ralbidv 2749 . . . . . . . . . . 11  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( A. x  e.  W  A. n  e.  ( 0 ... ( # `
 x ) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
170166, 169anbi12d 693 . . . . . . . . . 10  |-  ( r  =  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ->  ( ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  <->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
171170elabg 3085 . . . . . . . . 9  |-  ( {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  e.  _V  ->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  <->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
172165, 171syl 16 . . . . . . . 8  |-  ( ph  ->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  <->  ( { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) ) )
17325, 161, 172mpbir2and 890 . . . . . . 7  |-  ( ph  ->  { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) }  e.  { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) } )
174 intss1 4067 . . . . . . 7  |-  ( {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) }  e.  {
r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  ->  |^|
{ r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  C_  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) } )
175173, 174syl 16 . . . . . 6  |-  ( ph  ->  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  (
0 ... ( # `  x
) ) A. a  e.  I  A. b  e.  2o  x r ( x splice  <. n ,  n ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  C_  {
<. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) ) } )
1763, 175syl5eqss 3394 . . . . 5  |-  ( ph  ->  .~  C_  { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } )
177176ssbrd 4256 . . . 4  |-  ( ph  ->  ( A  .~  C  ->  A { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C ) )
178177imp 420 . . 3  |-  ( (
ph  /\  A  .~  C )  ->  A { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C )
1791, 2efger 15355 . . . . . 6  |-  .~  Er  W
180 errel 6917 . . . . . 6  |-  (  .~  Er  W  ->  Rel  .~  )
181179, 180mp1i 12 . . . . 5  |-  ( ph  ->  Rel  .~  )
182 brrelex12 4918 . . . . 5  |-  ( ( Rel  .~  /\  A  .~  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
183181, 182sylan 459 . . . 4  |-  ( (
ph  /\  A  .~  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
184 preq12 3887 . . . . . . 7  |-  ( ( u  =  A  /\  v  =  C )  ->  { u ,  v }  =  { A ,  C } )
185184sseq1d 3377 . . . . . 6  |-  ( ( u  =  A  /\  v  =  C )  ->  ( { u ,  v }  C_  W  <->  { A ,  C }  C_  W ) )
186 coeq2 5034 . . . . . . . 8  |-  ( u  =  A  ->  ( T  o.  u )  =  ( T  o.  A ) )
187186oveq2d 6100 . . . . . . 7  |-  ( u  =  A  ->  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  A
) ) )
188 coeq2 5034 . . . . . . . 8  |-  ( v  =  C  ->  ( T  o.  v )  =  ( T  o.  C ) )
189188oveq2d 6100 . . . . . . 7  |-  ( v  =  C  ->  ( H  gsumg  ( T  o.  v
) )  =  ( H  gsumg  ( T  o.  C
) ) )
190187, 189eqeqan12d 2453 . . . . . 6  |-  ( ( u  =  A  /\  v  =  C )  ->  ( ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) )  <-> 
( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) )
191185, 190anbi12d 693 . . . . 5  |-  ( ( u  =  A  /\  v  =  C )  ->  ( ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u ) )  =  ( H  gsumg  ( T  o.  v ) ) )  <->  ( { A ,  C }  C_  W  /\  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) ) )
192191, 157brabga 4472 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A { <. u ,  v >.  |  ( { u ,  v }  C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C  <->  ( { A ,  C }  C_  W  /\  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) ) )
193183, 192syl 16 . . 3  |-  ( (
ph  /\  A  .~  C )  ->  ( A { <. u ,  v
>.  |  ( {
u ,  v } 
C_  W  /\  ( H  gsumg  ( T  o.  u
) )  =  ( H  gsumg  ( T  o.  v
) ) ) } C  <->  ( { A ,  C }  C_  W  /\  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) ) )
194178, 193mpbid 203 . 2  |-  ( (
ph  /\  A  .~  C )  ->  ( { A ,  C }  C_  W  /\  ( H 
gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) ) )
195194simprd 451 1  |-  ( (
ph  /\  A  .~  C )  ->  ( H  gsumg  ( T  o.  A
) )  =  ( H  gsumg  ( T  o.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ifcif 3741   {cpr 3817   <.cop 3819   <.cotp 3820   |^|cint 4052   class class class wbr 4215   {copab 4268    _I cid 4496    X. cxp 4879    o. ccom 4885   Rel wrel 4886   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1oc1o 6720   2oc2o 6721    Er wer 6905   0cc0 8995   NN0cn0 10226   ZZ>=cuz 10493   ...cfz 11048   #chash 11623  Word cword 11722   concat cconcat 11723   substr csubstr 11725   splice csplice 11726   <"cs2 11810   Basecbs 13474   +g cplusg 13534   0gc0g 13728    gsumg cgsu 13729   Mndcmnd 14689   Grpcgrp 14690   inv gcminusg 14691   ~FG cefg 15343
This theorem is referenced by:  frgpupf  15410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-splice 11732  df-s2 11817  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-gsum 13733  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-efg 15346
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