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Theorem frgpnabllem1 15161
Description: Lemma for frgpnabl 15163. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
frgpnabl.g  |-  G  =  (freeGrp `  I )
frgpnabl.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpnabl.r  |-  .~  =  ( ~FG  `  I )
frgpnabl.p  |-  .+  =  ( +g  `  G )
frgpnabl.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
frgpnabl.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
frgpnabl.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
frgpnabl.u  |-  U  =  (varFGrp `  I )
frgpnabl.i  |-  ( ph  ->  I  e.  _V )
frgpnabl.a  |-  ( ph  ->  A  e.  I )
frgpnabl.b  |-  ( ph  ->  B  e.  I )
Assertion
Ref Expression
frgpnabllem1  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
Distinct variable groups:    x, A    v, n, w, x, y, z, I    ph, x    x, 
.~ , y, z    x, B    n, W, v, w, x, y, z    x, G    n, M, v, w, x    x, T
Allowed substitution hints:    ph( y, z, w, v, n)    A( y, z, w, v, n)    B( y, z, w, v, n)    D( x, y, z, w, v, n)    .+ ( x, y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    U( x, y, z, w, v, n)    G( y,
z, w, v, n)    M( y, z)

Proof of Theorem frgpnabllem1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpnabl.a . . . . . . 7  |-  ( ph  ->  A  e.  I )
2 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
32prid1 3734 . . . . . . . 8  |-  (/)  e.  { (/)
,  1o }
4 df2o3 6492 . . . . . . . 8  |-  2o  =  { (/) ,  1o }
53, 4eleqtrri 2356 . . . . . . 7  |-  (/)  e.  2o
6 opelxpi 4721 . . . . . . 7  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  ->  <. A ,  (/) >.  e.  ( I  X.  2o ) )
71, 5, 6sylancl 643 . . . . . 6  |-  ( ph  -> 
<. A ,  (/) >.  e.  ( I  X.  2o ) )
8 frgpnabl.b . . . . . . 7  |-  ( ph  ->  B  e.  I )
9 opelxpi 4721 . . . . . . 7  |-  ( ( B  e.  I  /\  (/) 
e.  2o )  ->  <. B ,  (/) >.  e.  ( I  X.  2o ) )
108, 5, 9sylancl 643 . . . . . 6  |-  ( ph  -> 
<. B ,  (/) >.  e.  ( I  X.  2o ) )
117, 10s2cld 11519 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e. Word  ( I  X.  2o ) )
12 frgpnabl.w . . . . . 6  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
13 frgpnabl.i . . . . . . . 8  |-  ( ph  ->  I  e.  _V )
14 2on 6487 . . . . . . . 8  |-  2o  e.  On
15 xpexg 4800 . . . . . . . 8  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
1613, 14, 15sylancl 643 . . . . . . 7  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
17 wrdexg 11425 . . . . . . 7  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
18 fvi 5579 . . . . . . 7  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1916, 17, 183syl 18 . . . . . 6  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2012, 19syl5eq 2327 . . . . 5  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
2111, 20eleqtrrd 2360 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
22 1n0 6494 . . . . . . 7  |-  1o  =/=  (/)
23 2cn 9816 . . . . . . . . . . . . . 14  |-  2  e.  CC
2423addid2i 9000 . . . . . . . . . . . . 13  |-  ( 0  +  2 )  =  2
25 s2len 11537 . . . . . . . . . . . . 13  |-  ( # `  <" <. A ,  (/)
>. <. B ,  (/) >. "> )  =  2
2624, 25eqtr4i 2306 . . . . . . . . . . . 12  |-  ( 0  +  2 )  =  ( # `  <"
<. A ,  (/) >. <. B ,  (/)
>. "> )
27 frgpnabl.r . . . . . . . . . . . . . 14  |-  .~  =  ( ~FG  `  I )
28 frgpnabl.m . . . . . . . . . . . . . 14  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
29 frgpnabl.t . . . . . . . . . . . . . 14  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
3012, 27, 28, 29efgtlen 15035 . . . . . . . . . . . . 13  |-  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )  -> 
( # `  <" <. A ,  (/) >. <. B ,  (/) >. "> )  =  ( ( # `  x
)  +  2 ) )
3130adantll 694 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  W )  /\  <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  x ) )  ->  ( # `  <"
<. A ,  (/) >. <. B ,  (/)
>. "> )  =  ( ( # `  x
)  +  2 ) )
3226, 31syl5eq 2327 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  W )  /\  <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  x ) )  ->  ( 0  +  2 )  =  ( ( # `  x
)  +  2 ) )
3332ex 423 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  (
0  +  2 )  =  ( ( # `  x )  +  2 ) ) )
34 0cn 8831 . . . . . . . . . . . 12  |-  0  e.  CC
3534a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  W )  ->  0  e.  CC )
36 simpr 447 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  W )  ->  x  e.  W )
3712efgrcl 15024 . . . . . . . . . . . . . . . 16  |-  ( x  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
3837simprd 449 . . . . . . . . . . . . . . 15  |-  ( x  e.  W  ->  W  = Word  ( I  X.  2o ) )
3938adantl 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  W )  ->  W  = Word  ( I  X.  2o ) )
4036, 39eleqtrd 2359 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  W )  ->  x  e. Word  ( I  X.  2o ) )
41 lencl 11421 . . . . . . . . . . . . 13  |-  ( x  e. Word  ( I  X.  2o )  ->  ( # `  x )  e.  NN0 )
4240, 41syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  W )  ->  ( # `
 x )  e. 
NN0 )
4342nn0cnd 10020 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  W )  ->  ( # `
 x )  e.  CC )
4423a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  W )  ->  2  e.  CC )
4535, 43, 44addcan2d 9016 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  W )  ->  (
( 0  +  2 )  =  ( (
# `  x )  +  2 )  <->  0  =  ( # `  x ) ) )
4633, 45sylibd 205 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  0  =  ( # `  x
) ) )
4712, 27, 28, 29efgtf 15031 . . . . . . . . . . . . . . . . . 18  |-  ( (/)  e.  W  ->  ( ( T `  (/) )  =  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o )  |->  (
(/) splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  /\  ( T `  (/) ) : ( ( 0 ... ( # `
 (/) ) )  X.  ( I  X.  2o ) ) --> W ) )
4847adantl 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( ( T `  (/) )  =  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o )  |->  (
(/) splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  /\  ( T `  (/) ) : ( ( 0 ... ( # `
 (/) ) )  X.  ( I  X.  2o ) ) --> W ) )
4948simpld 445 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( T `  (/) )  =  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( (/) splice  <. a ,  a ,  <" b
( M `  b
) "> >. )
) )
5049rneqd 4906 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  (/)  e.  W
)  ->  ran  ( T `
 (/) )  =  ran  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o )  |->  (
(/) splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5150eleq2d 2350 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  (/) )  <->  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  (
a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( (/) splice  <. a ,  a ,  <" b
( M `  b
) "> >. )
) ) )
52 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( (/) splice  <. a ,  a ,  <" b
( M `  b
) "> >. )
)  =  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( (/) splice  <. a ,  a ,  <" b
( M `  b
) "> >. )
)
53 ovex 5883 . . . . . . . . . . . . . . . 16  |-  ( (/) splice  <.
a ,  a , 
<" b ( M `
 b ) "> >. )  e.  _V
5452, 53elrnmpt2 5957 . . . . . . . . . . . . . . 15  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( a  e.  ( 0 ... ( # `
 (/) ) ) ,  b  e.  ( I  X.  2o )  |->  (
(/) splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  <->  E. a  e.  ( 0 ... ( # `  (/) ) ) E. b  e.  ( I  X.  2o ) <" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( (/) splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) )
55 wrd0 11418 . . . . . . . . . . . . . . . . . . . . 21  |-  (/)  e. Word  (
I  X.  2o )
5655a1i 10 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  e. Word  ( I  X.  2o ) )
57 simprr 733 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
5828efgmf 15022 . . . . . . . . . . . . . . . . . . . . . . 23  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
5958ffvelrni 5664 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
6057, 59syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
6157, 60s2cld 11519 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
62 ccatlid 11434 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
6355, 62ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (/) concat  (/) )  =  (/)
6463oveq1i 5868 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
(/) concat 
(/) ) concat  (/) )  =  ( (/) concat  (/) )
6564, 63eqtr2i 2304 . . . . . . . . . . . . . . . . . . . . 21  |-  (/)  =  ( ( (/) concat  (/) ) concat  (/) )
6665a1i 10 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  =  ( (
(/) concat 
(/) ) concat  (/) ) )
67 simprl 732 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... ( # `  (/) ) ) )
68 hash0 11355 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( # `  (/) )  =  0
6968oveq2i 5869 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0 ... ( # `  (/) ) )  =  ( 0 ... 0 )
7067, 69syl6eleq 2373 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... 0 ) )
71 elfz1eq 10807 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  e.  ( 0 ... 0 )  ->  a  =  0 )
7270, 71syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  0 )
7372, 68syl6eqr 2333 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  (
# `  (/) ) )
7468oveq2i 5869 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  +  ( # `  (/) ) )  =  ( a  +  0 )
7572, 34syl6eqel 2371 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  CC )
7675addid1d 9012 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( a  +  0 )  =  a )
7774, 76syl5req 2328 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a  +  ( # `  (/) ) ) )
7856, 56, 56, 61, 66, 73, 77splval2 11472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( (/) splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  =  ( (
(/) concat  <" b ( M `  b ) "> ) concat  (/) ) )
79 ccatlid 11434 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  ( (/) concat  <" b ( M `
 b ) "> )  =  <" b ( M `  b ) "> )
8079oveq1d 5873 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  (
( (/) concat  <" b ( M `  b ) "> ) concat  (/) )  =  ( <" b
( M `  b
) "> concat  (/) ) )
81 ccatrid 11435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  ( <" b ( M `
 b ) "> concat  (/) )  =  <" b ( M `  b ) "> )
8280, 81eqtrd 2315 . . . . . . . . . . . . . . . . . . . 20  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  (
( (/) concat  <" b ( M `  b ) "> ) concat  (/) )  = 
<" b ( M `
 b ) "> )
8361, 82syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( (/) concat  <" b ( M `
 b ) "> ) concat  (/) )  = 
<" b ( M `
 b ) "> )
8478, 83eqtrd 2315 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( (/) splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  =  <" b
( M `  b
) "> )
8584eqeq2d 2294 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( (/) splice  <.
a ,  a , 
<" b ( M `
 b ) "> >. )  <->  <" <. A ,  (/) >. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )
)
861ad3antrrr 710 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  A  e.  I )
87 1on 6486 . . . . . . . . . . . . . . . . . . . 20  |-  1o  e.  On
8887a1i 10 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  1o  e.  On )
89 simpr 447 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )
9089fveq1d 5527 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  1 )  =  ( <" b
( M `  b
) "> `  1
) )
91 opex 4237 . . . . . . . . . . . . . . . . . . . . . 22  |-  <. B ,  (/)
>.  e.  _V
92 s2fv1 11536 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  1 )  =  <. B ,  (/) >.
)
9391, 92ax-mp 8 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  1 )  =  <. B ,  (/) >.
94 fvex 5539 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( M `
 b )  e. 
_V
95 s2fv1 11536 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( M `  b )  e.  _V  ->  ( <" b ( M `
 b ) "> `  1 )  =  ( M `  b ) )
9694, 95ax-mp 8 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" b ( M `
 b ) "> `  1 )  =  ( M `  b )
9790, 93, 963eqtr3g 2338 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  -> 
<. B ,  (/) >.  =  ( M `  b ) )
9889fveq1d 5527 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  ( <" b
( M `  b
) "> `  0
) )
99 opex 4237 . . . . . . . . . . . . . . . . . . . . . . 23  |-  <. A ,  (/)
>.  e.  _V
100 s2fv0 11535 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
10199, 100ax-mp 8 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
102 vex 2791 . . . . . . . . . . . . . . . . . . . . . . 23  |-  b  e. 
_V
103 s2fv0 11535 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  e.  _V  ->  ( <" b ( M `
 b ) "> `  0 )  =  b )
104102, 103ax-mp 8 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <" b ( M `
 b ) "> `  0 )  =  b
10598, 101, 1043eqtr3g 2338 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  -> 
<. A ,  (/) >.  =  b )
106105fveq2d 5529 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( M `  <. A ,  (/) >. )  =  ( M `  b ) )
10728efgmval 15021 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  -> 
( A M (/) )  =  <. A , 
( 1o  \  (/) ) >.
)
10886, 5, 107sylancl 643 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( A M (/) )  =  <. A , 
( 1o  \  (/) ) >.
)
109 df-ov 5861 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A M (/) )  =  ( M `  <. A ,  (/)
>. )
110 dif0 3524 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1o 
\  (/) )  =  1o
111110opeq2i 3800 . . . . . . . . . . . . . . . . . . . . 21  |-  <. A , 
( 1o  \  (/) ) >.  =  <. A ,  1o >.
112108, 109, 1113eqtr3g 2338 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( M `  <. A ,  (/) >. )  =  <. A ,  1o >. )
11397, 106, 1123eqtr2rd 2322 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  -> 
<. A ,  1o >.  = 
<. B ,  (/) >. )
114 opthg 4246 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  I  /\  1o  e.  On )  -> 
( <. A ,  1o >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  1o  =  (/) ) ) )
115114simplbda 607 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  I  /\  1o  e.  On )  /\  <. A ,  1o >.  =  <. B ,  (/) >.
)  ->  1o  =  (/) )
11686, 88, 113, 115syl21anc 1181 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  1o  =  (/) )
117116ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  =  <" b
( M `  b
) ">  ->  1o  =  (/) ) )
11885, 117sylbid 206 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( (/) splice  <.
a ,  a , 
<" b ( M `
 b ) "> >. )  ->  1o  =  (/) ) )
119118rexlimdvva 2674 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( E. a  e.  ( 0 ... ( # `  (/) ) ) E. b  e.  ( I  X.  2o )
<" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( (/) splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  ->  1o  =  (/) ) )
12054, 119syl5bi 208 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o )  |->  (
(/) splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)  ->  1o  =  (/) ) )
12151, 120sylbid 206 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  (/) )  ->  1o  =  (/) ) )
122121expimpd 586 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) )  ->  1o  =  (/) ) )
123 vex 2791 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
124 hasheq0 11353 . . . . . . . . . . . . . . . 16  |-  ( x  e.  _V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
125123, 124ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  =  0  <->  x  =  (/) )
126 eleq1 2343 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( x  e.  W  <->  (/)  e.  W
) )
127 fveq2 5525 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  (/)  ->  ( T `
 x )  =  ( T `  (/) ) )
128127rneqd 4906 . . . . . . . . . . . . . . . . 17  |-  ( x  =  (/)  ->  ran  ( T `  x )  =  ran  ( T `  (/) ) )
129128eleq2d 2350 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  <->  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) )
130126, 129anbi12d 691 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )  <->  ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) ) )
131125, 130sylbi 187 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  0  ->  (
( x  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  <->  ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) ) )
132131eqcoms 2286 . . . . . . . . . . . . 13  |-  ( 0  =  ( # `  x
)  ->  ( (
x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )  <->  ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) ) )
133132imbi1d 308 . . . . . . . . . . . 12  |-  ( 0  =  ( # `  x
)  ->  ( (
( x  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  ->  1o  =  (/) )  <->  ( ( (/)  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) )  ->  1o  =  (/) ) ) )
134122, 133syl5ibrcom 213 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  =  (
# `  x )  ->  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  ->  1o  =  (/) ) ) )
135134com23 72 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  ->  ( 0  =  ( # `  x
)  ->  1o  =  (/) ) ) )
136135expdimp 426 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  (
0  =  ( # `  x )  ->  1o  =  (/) ) ) )
13746, 136mpdd 36 . . . . . . . 8  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  1o  =  (/) ) )
138137necon3ad 2482 . . . . . . 7  |-  ( (
ph  /\  x  e.  W )  ->  ( 1o  =/=  (/)  ->  -.  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )
) )
13922, 138mpi 16 . . . . . 6  |-  ( (
ph  /\  x  e.  W )  ->  -.  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )
140139nrexdv 2646 . . . . 5  |-  ( ph  ->  -.  E. x  e.  W  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)
141 eliun 3909 . . . . 5  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  U_ x  e.  W  ran  ( T `  x
)  <->  E. x  e.  W  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )
142140, 141sylnibr 296 . . . 4  |-  ( ph  ->  -.  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  U_ x  e.  W  ran  ( T `
 x ) )
143 eldif 3162 . . . 4  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( W  \  U_ x  e.  W  ran  ( T `  x ) )  <->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  W  /\  -.  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  U_ x  e.  W  ran  ( T `
 x ) ) )
14421, 142, 143sylanbrc 645 . . 3  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( W 
\  U_ x  e.  W  ran  ( T `  x
) ) )
145 frgpnabl.d . . 3  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
146144, 145syl6eleqr 2374 . 2  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
147 df-s2 11498 . . . . 5  |-  <" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> )
14812, 27efger 15027 . . . . . . 7  |-  .~  Er  W
149148a1i 10 . . . . . 6  |-  ( ph  ->  .~  Er  W )
150149, 21erref 6680 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. A ,  (/) >. <. B ,  (/) >. "> )
151147, 150syl5eqbrr 4057 . . . 4  |-  ( ph  ->  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/) >. "> )  .~  <" <. A ,  (/)
>. <. B ,  (/) >. "> )
152 ovex 5883 . . . . . 6  |-  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> )  e. 
_V
153147, 152eqeltri 2353 . . . . 5  |-  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  _V
154153, 152elec 6699 . . . 4  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  [ ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/) >. "> ) ]  .~  <->  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/) >. "> )  .~  <" <. A ,  (/)
>. <. B ,  (/) >. "> )
155151, 154sylibr 203 . . 3  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  [ (
<" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> ) ]  .~  )
156 frgpnabl.u . . . . . . 7  |-  U  =  (varFGrp `  I )
15727, 156vrgpval 15076 . . . . . 6  |-  ( ( I  e.  _V  /\  A  e.  I )  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
15813, 1, 157syl2anc 642 . . . . 5  |-  ( ph  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
15927, 156vrgpval 15076 . . . . . 6  |-  ( ( I  e.  _V  /\  B  e.  I )  ->  ( U `  B
)  =  [ <"
<. B ,  (/) >. "> ]  .~  )
16013, 8, 159syl2anc 642 . . . . 5  |-  ( ph  ->  ( U `  B
)  =  [ <"
<. B ,  (/) >. "> ]  .~  )
161158, 160oveq12d 5876 . . . 4  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( [
<" <. A ,  (/) >. "> ]  .~  .+  [
<" <. B ,  (/) >. "> ]  .~  )
)
1627s1cld 11442 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e. Word  (
I  X.  2o ) )
163162, 20eleqtrrd 2360 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e.  W
)
16410s1cld 11442 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. ">  e. Word  (
I  X.  2o ) )
165164, 20eleqtrrd 2360 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. ">  e.  W
)
166 frgpnabl.g . . . . . 6  |-  G  =  (freeGrp `  I )
167 frgpnabl.p . . . . . 6  |-  .+  =  ( +g  `  G )
16812, 166, 27, 167frgpadd 15072 . . . . 5  |-  ( (
<" <. A ,  (/) >. ">  e.  W  /\  <" <. B ,  (/) >. ">  e.  W )  ->  ( [ <"
<. A ,  (/) >. "> ]  .~  .+  [ <"
<. B ,  (/) >. "> ]  .~  )  =  [
( <" <. A ,  (/)
>. "> concat  <" <. B ,  (/) >. "> ) ]  .~  )
169163, 165, 168syl2anc 642 . . . 4  |-  ( ph  ->  ( [ <" <. A ,  (/) >. "> ]  .~  .+ 
[ <" <. B ,  (/)
>. "> ]  .~  )  =  [ ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> ) ]  .~  )
170161, 169eqtrd 2315 . . 3  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ (
<" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> ) ]  .~  )
171155, 170eleqtrrd 2360 . 2  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
172 elin 3358 . 2  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( D  i^i  (
( U `  A
)  .+  ( U `  B ) ) )  <-> 
( <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) ) )
173146, 171, 172sylanbrc 645 1  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    \ cdif 3149    i^i cin 3151   (/)c0 3455   {cpr 3641   <.cop 3643   <.cotp 3644   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    _I cid 4304   Oncon0 4392    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   [cec 6658   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740   2c2 9795   NN0cn0 9965   ...cfz 10782   #chash 11337  Word cword 11403   concat cconcat 11404   <"cs1 11405   splice csplice 11407   <"cs2 11491   +g cplusg 13208   ~FG cefg 15015  freeGrpcfrgp 15016  varFGrpcvrgp 15017
This theorem is referenced by:  frgpnabllem2  15162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-imas 13411  df-divs 13412  df-mnd 14367  df-frmd 14471  df-efg 15018  df-frgp 15019  df-vrgp 15020
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