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Theorem frgpnabllem1 15486
Description: Lemma for frgpnabl 15488. (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 4341 . . . . . . . . 9  |-  (/)  e.  _V
32prid1 3914 . . . . . . . 8  |-  (/)  e.  { (/)
,  1o }
4 df2o3 6739 . . . . . . . 8  |-  2o  =  { (/) ,  1o }
53, 4eleqtrri 2511 . . . . . . 7  |-  (/)  e.  2o
6 opelxpi 4912 . . . . . . 7  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  ->  <. A ,  (/) >.  e.  ( I  X.  2o ) )
71, 5, 6sylancl 645 . . . . . 6  |-  ( ph  -> 
<. A ,  (/) >.  e.  ( I  X.  2o ) )
8 frgpnabl.b . . . . . . 7  |-  ( ph  ->  B  e.  I )
9 opelxpi 4912 . . . . . . 7  |-  ( ( B  e.  I  /\  (/) 
e.  2o )  ->  <. B ,  (/) >.  e.  ( I  X.  2o ) )
108, 5, 9sylancl 645 . . . . . 6  |-  ( ph  -> 
<. B ,  (/) >.  e.  ( I  X.  2o ) )
117, 10s2cld 11835 . . . . 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 6734 . . . . . . . 8  |-  2o  e.  On
15 xpexg 4991 . . . . . . . 8  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
1613, 14, 15sylancl 645 . . . . . . 7  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
17 wrdexg 11741 . . . . . . 7  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
18 fvi 5785 . . . . . . 7  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
1916, 17, 183syl 19 . . . . . 6  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2012, 19syl5eq 2482 . . . . 5  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
2111, 20eleqtrrd 2515 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
22 1n0 6741 . . . . . . 7  |-  1o  =/=  (/)
23 2cn 10072 . . . . . . . . . . . . . 14  |-  2  e.  CC
2423addid2i 9256 . . . . . . . . . . . . 13  |-  ( 0  +  2 )  =  2
25 s2len 11853 . . . . . . . . . . . . 13  |-  ( # `  <" <. A ,  (/)
>. <. B ,  (/) >. "> )  =  2
2624, 25eqtr4i 2461 . . . . . . . . . . . 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 15360 . . . . . . . . . . . . 13  |-  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )  -> 
( # `  <" <. A ,  (/) >. <. B ,  (/) >. "> )  =  ( ( # `  x
)  +  2 ) )
3130adantll 696 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  W )  /\  <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  x ) )  ->  ( # `  <"
<. A ,  (/) >. <. B ,  (/)
>. "> )  =  ( ( # `  x
)  +  2 ) )
3226, 31syl5eq 2482 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  W )  /\  <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  x ) )  ->  ( 0  +  2 )  =  ( ( # `  x
)  +  2 ) )
3332ex 425 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  (
0  +  2 )  =  ( ( # `  x )  +  2 ) ) )
34 0cn 9086 . . . . . . . . . . . 12  |-  0  e.  CC
3534a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  W )  ->  0  e.  CC )
36 simpr 449 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  W )  ->  x  e.  W )
3712efgrcl 15349 . . . . . . . . . . . . . . . 16  |-  ( x  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
3837simprd 451 . . . . . . . . . . . . . . 15  |-  ( x  e.  W  ->  W  = Word  ( I  X.  2o ) )
3938adantl 454 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  W )  ->  W  = Word  ( I  X.  2o ) )
4036, 39eleqtrd 2514 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  W )  ->  x  e. Word  ( I  X.  2o ) )
41 lencl 11737 . . . . . . . . . . . . 13  |-  ( x  e. Word  ( I  X.  2o )  ->  ( # `  x )  e.  NN0 )
4240, 41syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  W )  ->  ( # `
 x )  e. 
NN0 )
4342nn0cnd 10278 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  W )  ->  ( # `
 x )  e.  CC )
4423a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  W )  ->  2  e.  CC )
4535, 43, 44addcan2d 9272 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  W )  ->  (
( 0  +  2 )  =  ( (
# `  x )  +  2 )  <->  0  =  ( # `  x ) ) )
4633, 45sylibd 207 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  0  =  ( # `  x
) ) )
4712, 27, 28, 29efgtf 15356 . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( T `  (/) )  =  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( (/) splice  <. a ,  a ,  <" b
( M `  b
) "> >. )
) )
5049rneqd 5099 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  (/)  e.  W
)  ->  ran  ( T `
 (/) )  =  ran  ( a  e.  ( 0 ... ( # `  (/) ) ) ,  b  e.  ( I  X.  2o )  |->  (
(/) splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
) )
5150eleq2d 2505 . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . 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 6108 . . . . . . . . . . . . . . . 16  |-  ( (/) splice  <.
a ,  a , 
<" b ( M `
 b ) "> >. )  e.  _V
5452, 53elrnmpt2 6185 . . . . . . . . . . . . . . 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 11734 . . . . . . . . . . . . . . . . . . . . 21  |-  (/)  e. Word  (
I  X.  2o )
5655a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  e. Word  ( I  X.  2o ) )
57 simprr 735 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  b  e.  ( I  X.  2o ) )
5828efgmf 15347 . . . . . . . . . . . . . . . . . . . . . . 23  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
5958ffvelrni 5871 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  e.  ( I  X.  2o )  ->  ( M `
 b )  e.  ( I  X.  2o ) )
6057, 59syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( M `  b )  e.  ( I  X.  2o ) )
6157, 60s2cld 11835 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  <" b ( M `  b ) ">  e. Word  (
I  X.  2o ) )
62 ccatlid 11750 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (/)  e. Word  ( I  X.  2o )  ->  ( (/) concat  (/) )  =  (/) )
6355, 62ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (/) concat  (/) )  =  (/)
6463oveq1i 6093 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
(/) concat 
(/) ) concat  (/) )  =  ( (/) concat  (/) )
6564, 63eqtr2i 2459 . . . . . . . . . . . . . . . . . . . . 21  |-  (/)  =  ( ( (/) concat  (/) ) concat  (/) )
6665a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  (/)  =  ( (
(/) concat 
(/) ) concat  (/) ) )
67 simprl 734 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... ( # `  (/) ) ) )
68 hash0 11648 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( # `  (/) )  =  0
6968oveq2i 6094 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0 ... ( # `  (/) ) )  =  ( 0 ... 0 )
7067, 69syl6eleq 2528 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  ( 0 ... 0 ) )
71 elfz1eq 11070 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  e.  ( 0 ... 0 )  ->  a  =  0 )
7270, 71syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  0 )
7372, 68syl6eqr 2488 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  (
# `  (/) ) )
7468oveq2i 6094 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  +  ( # `  (/) ) )  =  ( a  +  0 )
7572, 34syl6eqel 2526 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  e.  CC )
7675addid1d 9268 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( a  +  0 )  =  a )
7774, 76syl5req 2483 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  a  =  ( a  +  ( # `  (/) ) ) )
7856, 56, 56, 61, 66, 73, 77splval2 11788 . . . . . . . . . . . . . . . . . . 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 11750 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  ( (/) concat  <" b ( M `
 b ) "> )  =  <" b ( M `  b ) "> )
8079oveq1d 6098 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  (
( (/) concat  <" b ( M `  b ) "> ) concat  (/) )  =  ( <" b
( M `  b
) "> concat  (/) ) )
81 ccatrid 11751 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  ( <" b ( M `
 b ) "> concat  (/) )  =  <" b ( M `  b ) "> )
8280, 81eqtrd 2470 . . . . . . . . . . . . . . . . . . . 20  |-  ( <" b ( M `
 b ) ">  e. Word  ( I  X.  2o )  ->  (
( (/) concat  <" b ( M `  b ) "> ) concat  (/) )  = 
<" b ( M `
 b ) "> )
8361, 82syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( ( (/) concat  <" b ( M `
 b ) "> ) concat  (/) )  = 
<" b ( M `
 b ) "> )
8478, 83eqtrd 2470 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( (/) splice  <. a ,  a ,  <" b ( M `  b ) "> >.
)  =  <" b
( M `  b
) "> )
8584eqeq2d 2449 . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  A  e.  I )
87 1on 6733 . . . . . . . . . . . . . . . . . . . 20  |-  1o  e.  On
8887a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  1o  e.  On )
89 simpr 449 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )
9089fveq1d 5732 . . . . . . . . . . . . . . . . . . . . 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 4429 . . . . . . . . . . . . . . . . . . . . . 22  |-  <. B ,  (/)
>.  e.  _V
92 s2fv1 11852 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  1 )  =  <. B ,  (/) >.
)
9391, 92ax-mp 8 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  1 )  =  <. B ,  (/) >.
94 fvex 5744 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( M `
 b )  e. 
_V
95 s2fv1 11852 . . . . . . . . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  -> 
<. B ,  (/) >.  =  ( M `  b ) )
9889fveq1d 5732 . . . . . . . . . . . . . . . . . . . . . 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 4429 . . . . . . . . . . . . . . . . . . . . . . 23  |-  <. A ,  (/)
>.  e.  _V
100 s2fv0 11851 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
10199, 100ax-mp 8 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
102 vex 2961 . . . . . . . . . . . . . . . . . . . . . . 23  |-  b  e. 
_V
103 s2fv0 11851 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  e.  _V  ->  ( <" b ( M `
 b ) "> `  0 )  =  b )
104102, 103ax-mp 8 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <" b ( M `
 b ) "> `  0 )  =  b
10598, 101, 1043eqtr3g 2493 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  -> 
<. A ,  (/) >.  =  b )
106105fveq2d 5734 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( M `  <. A ,  (/) >. )  =  ( M `  b ) )
10728efgmval 15346 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  -> 
( A M (/) )  =  <. A , 
( 1o  \  (/) ) >.
)
10886, 5, 107sylancl 645 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( A M (/) )  =  <. A , 
( 1o  \  (/) ) >.
)
109 df-ov 6086 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A M (/) )  =  ( M `  <. A ,  (/)
>. )
110 dif0 3700 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1o 
\  (/) )  =  1o
111110opeq2i 3990 . . . . . . . . . . . . . . . . . . . . 21  |-  <. A , 
( 1o  \  (/) ) >.  =  <. A ,  1o >.
112108, 109, 1113eqtr3g 2493 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  ( M `  <. A ,  (/) >. )  =  <. A ,  1o >. )
11397, 106, 1123eqtr2rd 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  -> 
<. A ,  1o >.  = 
<. B ,  (/) >. )
114 opthg 4438 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  I  /\  1o  e.  On )  -> 
( <. A ,  1o >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  1o  =  (/) ) ) )
115114simplbda 609 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  I  /\  1o  e.  On )  /\  <. A ,  1o >.  =  <. B ,  (/) >.
)  ->  1o  =  (/) )
11686, 88, 113, 115syl21anc 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  (/) 
e.  W )  /\  ( a  e.  ( 0 ... ( # `  (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" b
( M `  b
) "> )  ->  1o  =  (/) )
117116ex 425 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  =  <" b
( M `  b
) ">  ->  1o  =  (/) ) )
11885, 117sylbid 208 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (/)  e.  W
)  /\  ( a  e.  ( 0 ... ( # `
 (/) ) )  /\  b  e.  ( I  X.  2o ) ) )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( (/) splice  <.
a ,  a , 
<" b ( M `
 b ) "> >. )  ->  1o  =  (/) ) )
119118rexlimdvva 2839 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . 13  |-  ( (
ph  /\  (/)  e.  W
)  ->  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  ran  ( T `  (/) )  ->  1o  =  (/) ) )
122121expimpd 588 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) )  ->  1o  =  (/) ) )
123 vex 2961 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
124 hasheq0 11646 . . . . . . . . . . . . . . . 16  |-  ( x  e.  _V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
125123, 124ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  =  0  <->  x  =  (/) )
126 eleq1 2498 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( x  e.  W  <->  (/)  e.  W
) )
127 fveq2 5730 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  (/)  ->  ( T `
 x )  =  ( T `  (/) ) )
128127rneqd 5099 . . . . . . . . . . . . . . . . 17  |-  ( x  =  (/)  ->  ran  ( T `  x )  =  ran  ( T `  (/) ) )
129128eleq2d 2505 . . . . . . . . . . . . . . . 16  |-  ( x  =  (/)  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  <->  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) )
130126, 129anbi12d 693 . . . . . . . . . . . . . . 15  |-  ( x  =  (/)  ->  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )  <->  ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) ) )
131125, 130sylbi 189 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  0  ->  (
( x  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  <->  ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) ) )
132131eqcoms 2441 . . . . . . . . . . . . 13  |-  ( 0  =  ( # `  x
)  ->  ( (
x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )  <->  ( (/)  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) ) ) )
133132imbi1d 310 . . . . . . . . . . . 12  |-  ( 0  =  ( # `  x
)  ->  ( (
( x  e.  W  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  ->  1o  =  (/) )  <->  ( ( (/)  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  (/) ) )  ->  1o  =  (/) ) ) )
134122, 133syl5ibrcom 215 . . . . . . . . . . 11  |-  ( ph  ->  ( 0  =  (
# `  x )  ->  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  ->  1o  =  (/) ) ) )
135134com23 75 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e.  W  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)  ->  ( 0  =  ( # `  x
)  ->  1o  =  (/) ) ) )
136135expdimp 428 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  (
0  =  ( # `  x )  ->  1o  =  (/) ) ) )
13746, 136mpdd 39 . . . . . . . 8  |-  ( (
ph  /\  x  e.  W )  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )  ->  1o  =  (/) ) )
138137necon3ad 2639 . . . . . . 7  |-  ( (
ph  /\  x  e.  W )  ->  ( 1o  =/=  (/)  ->  -.  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x )
) )
13922, 138mpi 17 . . . . . 6  |-  ( (
ph  /\  x  e.  W )  ->  -.  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )
140139nrexdv 2811 . . . . 5  |-  ( ph  ->  -.  E. x  e.  W  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ran  ( T `  x )
)
141 eliun 4099 . . . . 5  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  U_ x  e.  W  ran  ( T `  x
)  <->  E. x  e.  W  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ran  ( T `  x ) )
142140, 141sylnibr 298 . . . 4  |-  ( ph  ->  -.  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  U_ x  e.  W  ran  ( T `
 x ) )
14321, 142eldifd 3333 . . 3  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( W 
\  U_ x  e.  W  ran  ( T `  x
) ) )
144 frgpnabl.d . . 3  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
145143, 144syl6eleqr 2529 . 2  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
146 df-s2 11814 . . . . 5  |-  <" <. A ,  (/) >. <. B ,  (/) >. ">  =  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> )
14712, 27efger 15352 . . . . . . 7  |-  .~  Er  W
148147a1i 11 . . . . . 6  |-  ( ph  ->  .~  Er  W )
149148, 21erref 6927 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. A ,  (/) >. <. B ,  (/) >. "> )
150146, 149syl5eqbrr 4248 . . . 4  |-  ( ph  ->  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/) >. "> )  .~  <" <. A ,  (/)
>. <. B ,  (/) >. "> )
151 ovex 6108 . . . . . 6  |-  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> )  e. 
_V
152146, 151eqeltri 2508 . . . . 5  |-  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  _V
153152, 151elec 6946 . . . 4  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  [ ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/) >. "> ) ]  .~  <->  ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/) >. "> )  .~  <" <. A ,  (/)
>. <. B ,  (/) >. "> )
154150, 153sylibr 205 . . 3  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  [ (
<" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> ) ]  .~  )
155 frgpnabl.u . . . . . . 7  |-  U  =  (varFGrp `  I )
15627, 155vrgpval 15401 . . . . . 6  |-  ( ( I  e.  _V  /\  A  e.  I )  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
15713, 1, 156syl2anc 644 . . . . 5  |-  ( ph  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
15827, 155vrgpval 15401 . . . . . 6  |-  ( ( I  e.  _V  /\  B  e.  I )  ->  ( U `  B
)  =  [ <"
<. B ,  (/) >. "> ]  .~  )
15913, 8, 158syl2anc 644 . . . . 5  |-  ( ph  ->  ( U `  B
)  =  [ <"
<. B ,  (/) >. "> ]  .~  )
160157, 159oveq12d 6101 . . . 4  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( [
<" <. A ,  (/) >. "> ]  .~  .+  [
<" <. B ,  (/) >. "> ]  .~  )
)
1617s1cld 11758 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e. Word  (
I  X.  2o ) )
162161, 20eleqtrrd 2515 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e.  W
)
16310s1cld 11758 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. ">  e. Word  (
I  X.  2o ) )
164163, 20eleqtrrd 2515 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. ">  e.  W
)
165 frgpnabl.g . . . . . 6  |-  G  =  (freeGrp `  I )
166 frgpnabl.p . . . . . 6  |-  .+  =  ( +g  `  G )
16712, 165, 27, 166frgpadd 15397 . . . . 5  |-  ( (
<" <. A ,  (/) >. ">  e.  W  /\  <" <. B ,  (/) >. ">  e.  W )  ->  ( [ <"
<. A ,  (/) >. "> ]  .~  .+  [ <"
<. B ,  (/) >. "> ]  .~  )  =  [
( <" <. A ,  (/)
>. "> concat  <" <. B ,  (/) >. "> ) ]  .~  )
168162, 164, 167syl2anc 644 . . . 4  |-  ( ph  ->  ( [ <" <. A ,  (/) >. "> ]  .~  .+ 
[ <" <. B ,  (/)
>. "> ]  .~  )  =  [ ( <" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> ) ]  .~  )
169160, 168eqtrd 2470 . . 3  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ (
<" <. A ,  (/) >. "> concat  <" <. B ,  (/)
>. "> ) ]  .~  )
170154, 169eleqtrrd 2515 . 2  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
171 elin 3532 . 2  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( D  i^i  (
( U `  A
)  .+  ( U `  B ) ) )  <-> 
( <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) ) )
172145, 170, 171sylanbrc 647 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   _Vcvv 2958    \ cdif 3319    i^i cin 3321   (/)c0 3630   {cpr 3817   <.cop 3819   <.cotp 3820   U_ciun 4095   class class class wbr 4214    e. cmpt 4268    _I cid 4495   Oncon0 4583    X. cxp 4878   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720    Er wer 6904   [cec 6905   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995   2c2 10051   NN0cn0 10223   ...cfz 11045   #chash 11620  Word cword 11719   concat cconcat 11720   <"cs1 11721   splice csplice 11723   <"cs2 11807   +g cplusg 13531   ~FG cefg 15340  freeGrpcfrgp 15341  varFGrpcvrgp 15342
This theorem is referenced by:  frgpnabllem2  15487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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-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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-s2 11814  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-imas 13736  df-divs 13737  df-mnd 14692  df-frmd 14796  df-efg 15343  df-frgp 15344  df-vrgp 15345
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