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Theorem frgpup3lem 17437
Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
frgpup.b  |-  B  =  ( Base `  H
)
frgpup.n  |-  N  =  ( invg `  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 )
frgpup.g  |-  G  =  (freeGrp `  I )
frgpup.x  |-  X  =  ( Base `  G
)
frgpup.e  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
frgpup.u  |-  U  =  (varFGrp `  I )
frgpup3.k  |-  ( ph  ->  K  e.  ( G 
GrpHom  H ) )
frgpup3.e  |-  ( ph  ->  ( K  o.  U
)  =  F )
Assertion
Ref Expression
frgpup3lem  |-  ( ph  ->  K  =  E )
Distinct variable groups:    y, g,
z    g, H    y, F, z    y, N, z    B, g, y, z    T, g    .~ , g    ph, g, y, z    y, I, z   
g, W
Allowed substitution hints:    .~ ( y, z)    T( y, z)    U( y, z, g)    E( y, z, g)    F( g)    G( y, z, g)    H( y, z)    I( g)    K( y, z, g)    N( g)    V( y, z, g)    W( y, z)    X( y, z, g)

Proof of Theorem frgpup3lem
Dummy variables  a 
t  n  i  j  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup3.k . . 3  |-  ( ph  ->  K  e.  ( G 
GrpHom  H ) )
2 frgpup.x . . . 4  |-  X  =  ( Base `  G
)
3 frgpup.b . . . 4  |-  B  =  ( Base `  H
)
42, 3ghmf 16897 . . 3  |-  ( K  e.  ( G  GrpHom  H )  ->  K : X
--> B )
5 ffn 5710 . . 3  |-  ( K : X --> B  ->  K  Fn  X )
61, 4, 53syl 18 . 2  |-  ( ph  ->  K  Fn  X )
7 frgpup.n . . . 4  |-  N  =  ( invg `  H )
8 frgpup.t . . . 4  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
9 frgpup.h . . . 4  |-  ( ph  ->  H  e.  Grp )
10 frgpup.i . . . 4  |-  ( ph  ->  I  e.  V )
11 frgpup.a . . . 4  |-  ( ph  ->  F : I --> B )
12 frgpup.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
13 frgpup.r . . . 4  |-  .~  =  ( ~FG  `  I )
14 frgpup.g . . . 4  |-  G  =  (freeGrp `  I )
15 frgpup.e . . . 4  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
163, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15frgpup1 17435 . . 3  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
172, 3ghmf 16897 . . 3  |-  ( E  e.  ( G  GrpHom  H )  ->  E : X
--> B )
18 ffn 5710 . . 3  |-  ( E : X --> B  ->  E  Fn  X )
1916, 17, 183syl 18 . 2  |-  ( ph  ->  E  Fn  X )
20 eqid 2451 . . . . . . . . 9  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
2114, 20, 13frgpval 17418 . . . . . . . 8  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
2210, 21syl 17 . . . . . . 7  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
23 2on 7176 . . . . . . . . . . 11  |-  2o  e.  On
24 xpexg 6580 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
2510, 23, 24sylancl 673 . . . . . . . . . 10  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
26 wrdexg 12676 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
27 fvi 5905 . . . . . . . . . 10  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
2825, 26, 273syl 18 . . . . . . . . 9  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2912, 28syl5eq 2497 . . . . . . . 8  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
30 eqid 2451 . . . . . . . . . 10  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
3120, 30frmdbas 16646 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
3225, 31syl 17 . . . . . . . 8  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
3329, 32eqtr4d 2488 . . . . . . 7  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
34 fvex 5857 . . . . . . . . 9  |-  ( ~FG  `  I
)  e.  _V
3513, 34eqeltri 2525 . . . . . . . 8  |-  .~  e.  _V
3635a1i 11 . . . . . . 7  |-  ( ph  ->  .~  e.  _V )
37 fvex 5857 . . . . . . . 8  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3837a1i 11 . . . . . . 7  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3922, 33, 36, 38qusbas 15461 . . . . . 6  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
4039, 2syl6reqr 2504 . . . . 5  |-  ( ph  ->  X  =  ( W /.  .~  ) )
41 eqimss 3451 . . . . 5  |-  ( X  =  ( W /.  .~  )  ->  X  C_  ( W /.  .~  ) )
4240, 41syl 17 . . . 4  |-  ( ph  ->  X  C_  ( W /.  .~  ) )
4342sselda 3399 . . 3  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
44 eqid 2451 . . . 4  |-  ( W /.  .~  )  =  ( W /.  .~  )
45 fveq2 5847 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( K `  [ t ]  .~  )  =  ( K `  a ) )
46 fveq2 5847 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
4745, 46eqeq12d 2466 . . . 4  |-  ( [ t ]  .~  =  a  ->  ( ( K `
 [ t ]  .~  )  =  ( E `  [ t ]  .~  )  <->  ( K `  a )  =  ( E `  a ) ) )
48 simpr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  W )  ->  t  e.  W )
4929adantr 471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  W )  ->  W  = Word  ( I  X.  2o ) )
5048, 49eleqtrd 2531 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  t  e. Word  ( I  X.  2o ) )
51 wrdf 12670 . . . . . . . . . . . . 13  |-  ( t  e. Word  ( I  X.  2o )  ->  t : ( 0..^ ( # `  t ) ) --> ( I  X.  2o ) )
5250, 51syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  t : ( 0..^ (
# `  t )
) --> ( I  X.  2o ) )
5352ffvelrnda 6005 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( t `  n )  e.  ( I  X.  2o ) )
54 elxp2 4829 . . . . . . . . . . 11  |-  ( ( t `  n )  e.  ( I  X.  2o )  <->  E. i  e.  I  E. j  e.  2o  ( t `  n
)  =  <. i ,  j >. )
5553, 54sylib 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  E. i  e.  I  E. j  e.  2o  ( t `  n
)  =  <. i ,  j >. )
56 fveq2 5847 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( F `  y )  =  ( F `  i ) )
5756fveq2d 5851 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  i )
) )
5856, 57ifeq12d 3868 . . . . . . . . . . . . . . . 16  |-  ( y  =  i  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  if ( z  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
59 eqeq1 2455 . . . . . . . . . . . . . . . . 17  |-  ( z  =  j  ->  (
z  =  (/)  <->  j  =  (/) ) )
6059ifbid 3870 . . . . . . . . . . . . . . . 16  |-  ( z  =  j  ->  if ( z  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  if ( j  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
61 fvex 5857 . . . . . . . . . . . . . . . . 17  |-  ( F `
 i )  e. 
_V
62 fvex 5857 . . . . . . . . . . . . . . . . 17  |-  ( N `
 ( F `  i ) )  e. 
_V
6361, 62ifex 3916 . . . . . . . . . . . . . . . 16  |-  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  e.  _V
6458, 60, 8, 63ovmpt2 6419 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  I  /\  j  e.  2o )  ->  ( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
6564adantl 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
66 elpri 3952 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  { (/) ,  1o }  ->  ( j  =  (/)  \/  j  =  1o ) )
67 df2o3 7181 . . . . . . . . . . . . . . . . 17  |-  2o  =  { (/) ,  1o }
6866, 67eleq2s 2547 . . . . . . . . . . . . . . . 16  |-  ( j  e.  2o  ->  (
j  =  (/)  \/  j  =  1o ) )
69 frgpup3.e . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( K  o.  U
)  =  F )
7069adantr 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( K  o.  U )  =  F )
7170fveq1d 5849 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( F `  i ) )
72 frgpup.u . . . . . . . . . . . . . . . . . . . . . . 23  |-  U  =  (varFGrp `  I )
7313, 72, 14, 2vrgpf 17428 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( I  e.  V  ->  U : I --> X )
7410, 73syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  U : I --> X )
75 fvco3 5925 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U : I --> X  /\  i  e.  I )  ->  ( ( K  o.  U ) `  i
)  =  ( K `
 ( U `  i ) ) )
7674, 75sylan 478 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( K `  ( U `  i ) ) )
7771, 76eqtr3d 2487 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( F `  i )  =  ( K `  ( U `  i ) ) )
7877adantr 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( F `  i )  =  ( K `  ( U `  i ) ) )
79 iftrue 3854 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  (/)  ->  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( F `  i ) )
8079adantl 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( F `  i ) )
81 simpr 467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  j  =  (/) )
8281opeq2d 4142 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <. i ,  j >.  =  <. i ,  (/) >. )
8382s1eqd 12737 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <" <. i ,  j >. ">  =  <" <. i ,  (/) >. "> )
8483eceq1d 7386 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  (/) >. "> ]  .~  )
8513, 72vrgpval 17427 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( U `  i
)  =  [ <"
<. i ,  (/) >. "> ]  .~  )
8610, 85sylan 478 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8786adantr 471 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8884, 87eqtr4d 2488 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( U `  i ) )
8988fveq2d 5851 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( U `
 i ) ) )
9078, 80, 893eqtr4d 2495 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
9177fveq2d 5851 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( N `  ( K `  ( U `  i ) ) ) )
921adantr 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  K  e.  ( G  GrpHom  H ) )
9374ffvelrnda 6005 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  e.  X )
94 eqid 2451 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( invg `  G )  =  ( invg `  G )
952, 94, 7ghminv 16900 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( K  e.  ( G 
GrpHom  H )  /\  ( U `  i )  e.  X )  ->  ( K `  ( ( invg `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9692, 93, 95syl2anc 671 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( K `  ( ( invg `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9791, 96eqtr4d 2488 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( invg `  G ) `  ( U `  i )
) ) )
9897adantr 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( invg `  G ) `  ( U `  i )
) ) )
99 1n0 7183 . . . . . . . . . . . . . . . . . . . 20  |-  1o  =/=  (/)
100 simpr 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =  1o )
101100neeq1d 2682 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
j  =/=  (/)  <->  1o  =/=  (/) ) )
10299, 101mpbiri 241 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =/=  (/) )
103 ifnefalse 3860 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =/=  (/)  ->  if (
j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( N `  ( F `
 i ) ) )
104102, 103syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( N `  ( F `  i ) ) )
105100opeq2d 4142 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <. i ,  j >.  =  <. i ,  1o >. )
106105s1eqd 12737 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <" <. i ,  j >. ">  =  <" <. i ,  1o >. "> )
107106eceq1d 7386 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  1o >. "> ]  .~  )
10813, 72, 14, 94vrgpinv 17429 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( ( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
10910, 108sylan 478 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  (
( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
110109adantr 471 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
111107, 110eqtr4d 2488 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( ( invg `  G ) `  ( U `  i )
) )
112111fveq2d 5851 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( ( invg `  G
) `  ( U `  i ) ) ) )
11398, 104, 1123eqtr4d 2495 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11490, 113jaodan 799 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  I )  /\  (
j  =  (/)  \/  j  =  1o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11568, 114sylan2 481 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  I )  /\  j  e.  2o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
116115anasss 657 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11765, 116eqtrd 2485 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  ( K `
 [ <" <. i ,  j >. "> ]  .~  ) )
118 fveq2 5847 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( T `  <. i ,  j >. )
)
119 df-ov 6278 . . . . . . . . . . . . . . 15  |-  ( i T j )  =  ( T `  <. i ,  j >. )
120118, 119syl6eqr 2503 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( i T j ) )
121 s1eq 12736 . . . . . . . . . . . . . . . 16  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  <" ( t `
 n ) ">  =  <" <. i ,  j >. "> )
122121eceq1d 7386 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  [ <" (
t `  n ) "> ]  .~  =  [ <" <. i ,  j >. "> ]  .~  )
123122fveq2d 5851 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( K `  [ <" ( t `
 n ) "> ]  .~  )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
124120, 123eqeq12d 2466 . . . . . . . . . . . . 13  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( ( T `
 ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  )  <->  ( i T j )  =  ( K `  [ <" <. i ,  j
>. "> ]  .~  ) ) )
125117, 124syl5ibrcom 230 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
126125rexlimdvva 2858 . . . . . . . . . . 11  |-  ( ph  ->  ( E. i  e.  I  E. j  e.  2o  ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
127126ad2antrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( E. i  e.  I  E. j  e.  2o  ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
12855, 127mpd 15 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( T `  ( t `  n
) )  =  ( K `  [ <" ( t `  n
) "> ]  .~  ) )
129128mpteq2dva 4460 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
n  e.  ( 0..^ ( # `  t
) )  |->  ( T `
 ( t `  n ) ) )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( K `  [ <" ( t `  n
) "> ]  .~  ) ) )
1303, 7, 8, 9, 10, 11frgpuptf 17430 . . . . . . . . . 10  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
131130adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  T : ( I  X.  2o ) --> B )
132 fcompt 6043 . . . . . . . . 9  |-  ( ( T : ( I  X.  2o ) --> B  /\  t : ( 0..^ ( # `  t
) ) --> ( I  X.  2o ) )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t
) )  |->  ( T `
 ( t `  n ) ) ) )
133131, 52, 132syl2anc 671 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( T `  ( t `
 n ) ) ) )
13453s1cld 12739 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e. Word  ( I  X.  2o ) )
13529ad2antrr 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  W  = Word  (
I  X.  2o ) )
136134, 135eleqtrrd 2532 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e.  W )
13714, 13, 12, 2frgpeccl 17421 . . . . . . . . . 10  |-  ( <" ( t `  n ) ">  e.  W  ->  [ <" ( t `  n
) "> ]  .~  e.  X )
138136, 137syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  [ <" (
t `  n ) "> ]  .~  e.  X )
13952feqmptd 5900 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  t  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( t `  n ) ) )
14010adantr 471 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  I  e.  V )
141140, 23, 24sylancl 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  (
I  X.  2o )  e.  _V )
142 eqid 2451 . . . . . . . . . . . . 13  |-  (varFMnd `  (
I  X.  2o ) )  =  (varFMnd `  (
I  X.  2o ) )
143142vrmdfval 16650 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
144141, 143syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
145 s1eq 12736 . . . . . . . . . . 11  |-  ( w  =  ( t `  n )  ->  <" w ">  =  <" (
t `  n ) "> )
14653, 139, 144, 145fmptco 6040 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  <" ( t `
 n ) "> ) )
147 eqidd 2452 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  ) )
148 eceq1 7385 . . . . . . . . . 10  |-  ( w  =  <" ( t `
 n ) ">  ->  [ w ]  .~  =  [ <" ( t `  n
) "> ]  .~  )
149136, 146, 147, 148fmptco 6040 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  [ <" (
t `  n ) "> ]  .~  )
)
1501adantr 471 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G  GrpHom  H ) )
151150, 4syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  K : X --> B )
152151feqmptd 5900 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  K  =  ( w  e.  X  |->  ( K `  w ) ) )
153 fveq2 5847 . . . . . . . . 9  |-  ( w  =  [ <" (
t `  n ) "> ]  .~  ->  ( K `  w )  =  ( K `  [ <" ( t `
 n ) "> ]  .~  )
)
154138, 149, 152, 153fmptco 6040 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  ( K  o.  ( (
w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  ( n  e.  ( 0..^ ( # `  t
) )  |->  ( K `
 [ <" (
t `  n ) "> ]  .~  )
) )
155129, 133, 1543eqtr4d 2495 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( K  o.  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )
156155oveq2d 6291 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( H  gsumg  ( T  o.  t
) )  =  ( H  gsumg  ( K  o.  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) ) )
1573, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15frgpupval 17434 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( H  gsumg  ( T  o.  t
) ) )
158 ghmmhm 16903 . . . . . . . 8  |-  ( K  e.  ( G  GrpHom  H )  ->  K  e.  ( G MndHom  H ) )
159150, 158syl 17 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G MndHom  H ) )
160142vrmdf 16652 . . . . . . . . . . 11  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
161141, 160syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
16249feq3d 5697 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W  <->  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) ) )
163161, 162mpbird 240 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) --> W )
164 wrdco 12926 . . . . . . . . 9  |-  ( ( t  e. Word  ( I  X.  2o )  /\  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
16550, 163, 164syl2anc 671 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
16633adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
167166mpteq1d 4455 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
)
168 eqid 2451 . . . . . . . . . . . . 13  |-  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
16920, 30, 14, 13, 168frgpmhm 17425 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
170140, 169syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
171167, 170eqeltrd 2529 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G ) )
17230, 2mhmf 16597 . . . . . . . . . 10  |-  ( ( w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G )  -> 
( w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
173171, 172syl 17 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
174166feq2d 5696 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) : W --> X  <->  ( w  e.  W  |->  [ w ]  .~  ) : (
Base `  (freeMnd `  (
I  X.  2o ) ) ) --> X ) )
175173, 174mpbird 240 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : W --> X )
176 wrdco 12926 . . . . . . . 8  |-  ( ( ( (varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W  /\  ( w  e.  W  |->  [ w ]  .~  ) : W --> X )  ->  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) )  e. Word  X
)
177165, 175, 176syl2anc 671 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  e. Word  X )
1782gsumwmhm 16639 . . . . . . 7  |-  ( ( K  e.  ( G MndHom  H )  /\  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  e. Word  X )  ->  ( K `  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )  =  ( H  gsumg  ( K  o.  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) ) )
179159, 177, 178syl2anc 671 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) )  =  ( H  gsumg  ( K  o.  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) ) )
180156, 157, 1793eqtr4d 2495 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( K `  ( G 
gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) ) )
18120, 142frmdgsum 16656 . . . . . . . . 9  |-  ( ( ( I  X.  2o )  e.  _V  /\  t  e. Word  ( I  X.  2o ) )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
182141, 50, 181syl2anc 671 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
183182fveq2d 5851 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) `  ( (freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  ( ( w  e.  W  |->  [ w ]  .~  ) `  t ) )
184 wrdco 12926 . . . . . . . . . 10  |-  ( ( t  e. Word  ( I  X.  2o )  /\  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word Word  ( I  X.  2o ) )
18550, 161, 184syl2anc 671 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word Word  ( I  X.  2o ) )
18632adantr 471 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
187 wrdeq 12686 . . . . . . . . . 10  |-  ( (
Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
188186, 187syl 17 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
189185, 188eleqtrrd 2532 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
19030gsumwmhm 16639 . . . . . . . 8  |-  ( ( ( w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G )  /\  ( (varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )  ->  ( (
w  e.  W  |->  [ w ]  .~  ) `  ( (freeMnd `  (
I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )
191171, 189, 190syl2anc 671 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) `  ( (freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )
19212, 13efger 17378 . . . . . . . . 9  |-  .~  Er  W
193192a1i 11 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  .~  Er  W )
194 fvex 5857 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
19512, 194eqeltri 2525 . . . . . . . . 9  |-  W  e. 
_V
196195a1i 11 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  W  e.  _V )
197 eqid 2451 . . . . . . . 8  |-  ( w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  )
198193, 196, 197divsfval 15463 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) `  t )  =  [ t ]  .~  )
199183, 191, 1983eqtr3d 2493 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  [ t ]  .~  )
200199fveq2d 5851 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) )  =  ( K `  [
t ]  .~  )
)
201180, 200eqtr2d 2486 . . . 4  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  [ t ]  .~  )  =  ( E `  [ t ]  .~  ) )
20244, 47, 201ectocld 7416 . . 3  |-  ( (
ph  /\  a  e.  ( W /.  .~  )
)  ->  ( K `  a )  =  ( E `  a ) )
20343, 202syldan 477 . 2  |-  ( (
ph  /\  a  e.  X )  ->  ( K `  a )  =  ( E `  a ) )
2046, 19, 203eqfnfvd 5962 1  |-  ( ph  ->  K  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1447    e. wcel 1890    =/= wne 2621   E.wrex 2737   _Vcvv 3012    C_ wss 3371   (/)c0 3698   ifcif 3848   {cpr 3937   <.cop 3941    |-> cmpt 4432    _I cid 4721    X. cxp 4809   ran crn 4812    o. ccom 4815   Oncon0 5401    Fn wfn 5555   -->wf 5556   ` cfv 5560  (class class class)co 6275    |-> cmpt2 6277   1oc1o 7161   2oc2o 7162    Er wer 7346   [cec 7347   /.cqs 7348   0cc0 9525  ..^cfzo 11907   #chash 12508  Word cword 12650   <"cs1 12653   Basecbs 15131    gsumg cgsu 15349    /.s cqus 15414   MndHom cmhm 16590  freeMndcfrmd 16641  varFMndcvrmd 16642   Grpcgrp 16679   invgcminusg 16680    GrpHom cghm 16890   ~FG cefg 17366  freeGrpcfrgp 17367  varFGrpcvrgp 17368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570  ax-cnex 9581  ax-resscn 9582  ax-1cn 9583  ax-icn 9584  ax-addcl 9585  ax-addrcl 9586  ax-mulcl 9587  ax-mulrcl 9588  ax-mulcom 9589  ax-addass 9590  ax-mulass 9591  ax-distr 9592  ax-i2m1 9593  ax-1ne0 9594  ax-1rid 9595  ax-rnegex 9596  ax-rrecex 9597  ax-cnre 9598  ax-pre-lttri 9599  ax-pre-lttrn 9600  ax-pre-ltadd 9601  ax-pre-mulgt0 9602
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-pss 3387  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-tp 3940  df-op 3942  df-ot 3944  df-uni 4168  df-int 4204  df-iun 4249  df-iin 4250  df-br 4374  df-opab 4433  df-mpt 4434  df-tr 4469  df-eprel 4722  df-id 4726  df-po 4732  df-so 4733  df-fr 4770  df-we 4772  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-pred 5358  df-ord 5404  df-on 5405  df-lim 5406  df-suc 5407  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6680  df-1st 6780  df-2nd 6781  df-wrecs 7014  df-recs 7076  df-rdg 7114  df-1o 7168  df-2o 7169  df-oadd 7172  df-er 7349  df-ec 7351  df-qs 7355  df-map 7460  df-pm 7461  df-en 7556  df-dom 7557  df-sdom 7558  df-fin 7559  df-sup 7942  df-inf 7943  df-card 8359  df-cda 8584  df-pnf 9663  df-mnf 9664  df-xr 9665  df-ltxr 9666  df-le 9667  df-sub 9848  df-neg 9849  df-nn 10598  df-2 10656  df-3 10657  df-4 10658  df-5 10659  df-6 10660  df-7 10661  df-8 10662  df-9 10663  df-10 10664  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11775  df-fzo 11908  df-seq 12207  df-hash 12509  df-word 12658  df-lsw 12659  df-concat 12660  df-s1 12661  df-substr 12662  df-splice 12663  df-reverse 12664  df-s2 12942  df-struct 15133  df-ndx 15134  df-slot 15135  df-base 15136  df-sets 15137  df-ress 15138  df-plusg 15213  df-mulr 15214  df-sca 15216  df-vsca 15217  df-ip 15218  df-tset 15219  df-ple 15220  df-ds 15222  df-0g 15350  df-gsum 15351  df-imas 15417  df-qus 15419  df-mgm 16498  df-sgrp 16537  df-mnd 16547  df-mhm 16592  df-submnd 16593  df-frmd 16643  df-vrmd 16644  df-grp 16683  df-minusg 16684  df-ghm 16891  df-efg 17369  df-frgp 17370  df-vrgp 17371
This theorem is referenced by:  frgpup3  17438
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