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Theorem frgpup3lem 16279
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 15756 . . 3  |-  ( K  e.  ( G  GrpHom  H )  ->  K : X
--> B )
5 ffn 5564 . . 3  |-  ( K : X --> B  ->  K  Fn  X )
61, 4, 53syl 20 . 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 16277 . . 3  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
172, 3ghmf 15756 . . 3  |-  ( E  e.  ( G  GrpHom  H )  ->  E : X
--> B )
18 ffn 5564 . . 3  |-  ( E : X --> B  ->  E  Fn  X )
1916, 17, 183syl 20 . 2  |-  ( ph  ->  E  Fn  X )
20 eqid 2443 . . . . . . . . 9  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
2114, 20, 13frgpval 16260 . . . . . . . 8  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
2210, 21syl 16 . . . . . . 7  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
23 2on 6933 . . . . . . . . . . 11  |-  2o  e.  On
24 xpexg 6512 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
2510, 23, 24sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
26 wrdexg 12249 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
27 fvi 5753 . . . . . . . . . 10  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
2825, 26, 273syl 20 . . . . . . . . 9  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2912, 28syl5eq 2487 . . . . . . . 8  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
30 eqid 2443 . . . . . . . . . 10  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
3120, 30frmdbas 15535 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
3225, 31syl 16 . . . . . . . 8  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
3329, 32eqtr4d 2478 . . . . . . 7  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
34 fvex 5706 . . . . . . . . 9  |-  ( ~FG  `  I
)  e.  _V
3513, 34eqeltri 2513 . . . . . . . 8  |-  .~  e.  _V
3635a1i 11 . . . . . . 7  |-  ( ph  ->  .~  e.  _V )
37 fvex 5706 . . . . . . . 8  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3837a1i 11 . . . . . . 7  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3922, 33, 36, 38divsbas 14488 . . . . . 6  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
4039, 2syl6reqr 2494 . . . . 5  |-  ( ph  ->  X  =  ( W /.  .~  ) )
41 eqimss 3413 . . . . 5  |-  ( X  =  ( W /.  .~  )  ->  X  C_  ( W /.  .~  ) )
4240, 41syl 16 . . . 4  |-  ( ph  ->  X  C_  ( W /.  .~  ) )
4342sselda 3361 . . 3  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
44 eqid 2443 . . . 4  |-  ( W /.  .~  )  =  ( W /.  .~  )
45 fveq2 5696 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( K `  [ t ]  .~  )  =  ( K `  a ) )
46 fveq2 5696 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
4745, 46eqeq12d 2457 . . . 4  |-  ( [ t ]  .~  =  a  ->  ( ( K `
 [ t ]  .~  )  =  ( E `  [ t ]  .~  )  <->  ( K `  a )  =  ( E `  a ) ) )
48 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  W )  ->  t  e.  W )
4929adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  W )  ->  W  = Word  ( I  X.  2o ) )
5048, 49eleqtrd 2519 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  t  e. Word  ( I  X.  2o ) )
51 wrdf 12245 . . . . . . . . . . . . 13  |-  ( t  e. Word  ( I  X.  2o )  ->  t : ( 0..^ ( # `  t ) ) --> ( I  X.  2o ) )
5250, 51syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  t : ( 0..^ (
# `  t )
) --> ( I  X.  2o ) )
5352ffvelrnda 5848 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( t `  n )  e.  ( I  X.  2o ) )
54 elxp2 4863 . . . . . . . . . . 11  |-  ( ( t `  n )  e.  ( I  X.  2o )  <->  E. i  e.  I  E. j  e.  2o  ( t `  n
)  =  <. i ,  j >. )
5553, 54sylib 196 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  E. i  e.  I  E. j  e.  2o  ( t `  n
)  =  <. i ,  j >. )
56 fveq2 5696 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( F `  y )  =  ( F `  i ) )
5756fveq2d 5700 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  i )
) )
5856, 57ifeq12d 3814 . . . . . . . . . . . . . . . 16  |-  ( y  =  i  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  if ( z  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
59 eqeq1 2449 . . . . . . . . . . . . . . . . 17  |-  ( z  =  j  ->  (
z  =  (/)  <->  j  =  (/) ) )
6059ifbid 3816 . . . . . . . . . . . . . . . 16  |-  ( z  =  j  ->  if ( z  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  if ( j  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
61 fvex 5706 . . . . . . . . . . . . . . . . 17  |-  ( F `
 i )  e. 
_V
62 fvex 5706 . . . . . . . . . . . . . . . . 17  |-  ( N `
 ( F `  i ) )  e. 
_V
6361, 62ifex 3863 . . . . . . . . . . . . . . . 16  |-  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  e.  _V
6458, 60, 8, 63ovmpt2 6231 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  I  /\  j  e.  2o )  ->  ( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
6564adantl 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
66 elpri 3902 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  { (/) ,  1o }  ->  ( j  =  (/)  \/  j  =  1o ) )
67 df2o3 6938 . . . . . . . . . . . . . . . . 17  |-  2o  =  { (/) ,  1o }
6866, 67eleq2s 2535 . . . . . . . . . . . . . . . 16  |-  ( j  e.  2o  ->  (
j  =  (/)  \/  j  =  1o ) )
69 frgpup3.e . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( K  o.  U
)  =  F )
7069adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( K  o.  U )  =  F )
7170fveq1d 5698 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( F `  i ) )
72 frgpup.u . . . . . . . . . . . . . . . . . . . . . . 23  |-  U  =  (varFGrp `  I )
7313, 72, 14, 2vrgpf 16270 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( I  e.  V  ->  U : I --> X )
7410, 73syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  U : I --> X )
75 fvco3 5773 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U : I --> X  /\  i  e.  I )  ->  ( ( K  o.  U ) `  i
)  =  ( K `
 ( U `  i ) ) )
7674, 75sylan 471 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( K `  ( U `  i ) ) )
7771, 76eqtr3d 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( F `  i )  =  ( K `  ( U `  i ) ) )
7877adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( F `  i )  =  ( K `  ( U `  i ) ) )
79 iftrue 3802 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  (/)  ->  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( F `  i ) )
8079adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( F `  i ) )
81 simpr 461 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  j  =  (/) )
8281opeq2d 4071 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <. i ,  j >.  =  <. i ,  (/) >. )
8382s1eqd 12297 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <" <. i ,  j >. ">  =  <" <. i ,  (/) >. "> )
84 eceq1 7142 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" <. i ,  j
>. ">  =  <"
<. i ,  (/) >. ">  ->  [ <" <. i ,  j >. "> ]  .~  =  [ <"
<. i ,  (/) >. "> ]  .~  )
8583, 84syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  (/) >. "> ]  .~  )
8613, 72vrgpval 16269 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( U `  i
)  =  [ <"
<. i ,  (/) >. "> ]  .~  )
8710, 86sylan 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8887adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8985, 88eqtr4d 2478 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( U `  i ) )
9089fveq2d 5700 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( U `
 i ) ) )
9178, 80, 903eqtr4d 2485 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
9277fveq2d 5700 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( N `  ( K `  ( U `  i ) ) ) )
931adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  K  e.  ( G  GrpHom  H ) )
9474ffvelrnda 5848 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  e.  X )
95 eqid 2443 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( invg `  G )  =  ( invg `  G )
962, 95, 7ghminv 15759 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( K  e.  ( G 
GrpHom  H )  /\  ( U `  i )  e.  X )  ->  ( K `  ( ( invg `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9793, 94, 96syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( K `  ( ( invg `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9892, 97eqtr4d 2478 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( invg `  G ) `  ( U `  i )
) ) )
9998adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( invg `  G ) `  ( U `  i )
) ) )
100 1n0 6940 . . . . . . . . . . . . . . . . . . . 20  |-  1o  =/=  (/)
101 simpr 461 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =  1o )
102101neeq1d 2626 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
j  =/=  (/)  <->  1o  =/=  (/) ) )
103100, 102mpbiri 233 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =/=  (/) )
104 ifnefalse 3806 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =/=  (/)  ->  if (
j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( N `  ( F `
 i ) ) )
105103, 104syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( N `  ( F `  i ) ) )
106101opeq2d 4071 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <. i ,  j >.  =  <. i ,  1o >. )
107106s1eqd 12297 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <" <. i ,  j >. ">  =  <" <. i ,  1o >. "> )
108 eceq1 7142 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" <. i ,  j
>. ">  =  <"
<. i ,  1o >. ">  ->  [ <" <. i ,  j >. "> ]  .~  =  [ <"
<. i ,  1o >. "> ]  .~  )
109107, 108syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  1o >. "> ]  .~  )
11013, 72, 14, 95vrgpinv 16271 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( ( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
11110, 110sylan 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  (
( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
112111adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
113109, 112eqtr4d 2478 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( ( invg `  G ) `  ( U `  i )
) )
114113fveq2d 5700 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( ( invg `  G
) `  ( U `  i ) ) ) )
11599, 105, 1143eqtr4d 2485 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11691, 115jaodan 783 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  I )  /\  (
j  =  (/)  \/  j  =  1o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11768, 116sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  I )  /\  j  e.  2o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
118117anasss 647 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11965, 118eqtrd 2475 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  ( K `
 [ <" <. i ,  j >. "> ]  .~  ) )
120 fveq2 5696 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( T `  <. i ,  j >. )
)
121 df-ov 6099 . . . . . . . . . . . . . . 15  |-  ( i T j )  =  ( T `  <. i ,  j >. )
122120, 121syl6eqr 2493 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( i T j ) )
123 s1eq 12296 . . . . . . . . . . . . . . . 16  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  <" ( t `
 n ) ">  =  <" <. i ,  j >. "> )
124 eceq1 7142 . . . . . . . . . . . . . . . 16  |-  ( <" ( t `  n ) ">  =  <" <. i ,  j >. ">  ->  [ <" (
t `  n ) "> ]  .~  =  [ <" <. i ,  j >. "> ]  .~  )
125123, 124syl 16 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  [ <" (
t `  n ) "> ]  .~  =  [ <" <. i ,  j >. "> ]  .~  )
126125fveq2d 5700 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( K `  [ <" ( t `
 n ) "> ]  .~  )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
127122, 126eqeq12d 2457 . . . . . . . . . . . . 13  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( ( T `
 ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  )  <->  ( i T j )  =  ( K `  [ <" <. i ,  j
>. "> ]  .~  ) ) )
128119, 127syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
129128rexlimdvva 2853 . . . . . . . . . . 11  |-  ( ph  ->  ( E. i  e.  I  E. j  e.  2o  ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
130129ad2antrr 725 . . . . . . . . . 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 ) "> ]  .~  ) ) )
13155, 130mpd 15 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( T `  ( t `  n
) )  =  ( K `  [ <" ( t `  n
) "> ]  .~  ) )
132131mpteq2dva 4383 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
n  e.  ( 0..^ ( # `  t
) )  |->  ( T `
 ( t `  n ) ) )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( K `  [ <" ( t `  n
) "> ]  .~  ) ) )
1333, 7, 8, 9, 10, 11frgpuptf 16272 . . . . . . . . . 10  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
134133adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  T : ( I  X.  2o ) --> B )
135 fcompt 5884 . . . . . . . . 9  |-  ( ( T : ( I  X.  2o ) --> B  /\  t : ( 0..^ ( # `  t
) ) --> ( I  X.  2o ) )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t
) )  |->  ( T `
 ( t `  n ) ) ) )
136134, 52, 135syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( T `  ( t `
 n ) ) ) )
13753s1cld 12299 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e. Word  ( I  X.  2o ) )
13829ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  W  = Word  (
I  X.  2o ) )
139137, 138eleqtrrd 2520 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e.  W )
14014, 13, 12, 2frgpeccl 16263 . . . . . . . . . 10  |-  ( <" ( t `  n ) ">  e.  W  ->  [ <" ( t `  n
) "> ]  .~  e.  X )
141139, 140syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  [ <" (
t `  n ) "> ]  .~  e.  X )
14252feqmptd 5749 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  t  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( t `  n ) ) )
14310adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  I  e.  V )
144143, 23, 24sylancl 662 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  (
I  X.  2o )  e.  _V )
145 eqid 2443 . . . . . . . . . . . . 13  |-  (varFMnd `  (
I  X.  2o ) )  =  (varFMnd `  (
I  X.  2o ) )
146145vrmdfval 15539 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
147144, 146syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
148 s1eq 12296 . . . . . . . . . . 11  |-  ( w  =  ( t `  n )  ->  <" w ">  =  <" (
t `  n ) "> )
14953, 142, 147, 148fmptco 5881 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  <" ( t `
 n ) "> ) )
150 eqidd 2444 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  ) )
151 eceq1 7142 . . . . . . . . . 10  |-  ( w  =  <" ( t `
 n ) ">  ->  [ w ]  .~  =  [ <" ( t `  n
) "> ]  .~  )
152139, 149, 150, 151fmptco 5881 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  [ <" (
t `  n ) "> ]  .~  )
)
1531adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G  GrpHom  H ) )
154153, 4syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  K : X --> B )
155154feqmptd 5749 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  K  =  ( w  e.  X  |->  ( K `  w ) ) )
156 fveq2 5696 . . . . . . . . 9  |-  ( w  =  [ <" (
t `  n ) "> ]  .~  ->  ( K `  w )  =  ( K `  [ <" ( t `
 n ) "> ]  .~  )
)
157141, 152, 155, 156fmptco 5881 . . . . . . . 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 ) "> ]  .~  )
) )
158132, 136, 1573eqtr4d 2485 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( K  o.  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )
159158oveq2d 6112 . . . . . 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 ) ) ) ) )
1603, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15frgpupval 16276 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( H  gsumg  ( T  o.  t
) ) )
161 ghmmhm 15762 . . . . . . . 8  |-  ( K  e.  ( G  GrpHom  H )  ->  K  e.  ( G MndHom  H ) )
162153, 161syl 16 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G MndHom  H ) )
163145vrmdf 15541 . . . . . . . . . . 11  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
164144, 163syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
165 feq3 5549 . . . . . . . . . . 11  |-  ( W  = Word  ( I  X.  2o )  ->  ( (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W  <->  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) ) )
16649, 165syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W  <->  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) ) )
167164, 166mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) --> W )
168 wrdco 12464 . . . . . . . . 9  |-  ( ( t  e. Word  ( I  X.  2o )  /\  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
16950, 167, 168syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
17033adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
171170mpteq1d 4378 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
)
172 eqid 2443 . . . . . . . . . . . . 13  |-  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
17320, 30, 14, 13, 172frgpmhm 16267 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
174143, 173syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
175171, 174eqeltrd 2517 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G ) )
17630, 2mhmf 15474 . . . . . . . . . 10  |-  ( ( w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G )  -> 
( w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
177175, 176syl 16 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
178170feq2d 5552 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) : W --> X  <->  ( w  e.  W  |->  [ w ]  .~  ) : (
Base `  (freeMnd `  (
I  X.  2o ) ) ) --> X ) )
179177, 178mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : W --> X )
180 wrdco 12464 . . . . . . . 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
)
181169, 179, 180syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  e. Word  X )
1822gsumwmhm 15528 . . . . . . 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
) ) ) ) )
183162, 181, 182syl2anc 661 . . . . . 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 ) ) ) ) )
184159, 160, 1833eqtr4d 2485 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( K `  ( G 
gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) ) )
18520, 145frmdgsum 15545 . . . . . . . . 9  |-  ( ( ( I  X.  2o )  e.  _V  /\  t  e. Word  ( I  X.  2o ) )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
186144, 50, 185syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
187186fveq2d 5700 . . . . . . 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 ) )
188 wrdco 12464 . . . . . . . . . 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 ) )
18950, 164, 188syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word Word  ( I  X.  2o ) )
19032adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
191 wrdeq 12256 . . . . . . . . . 10  |-  ( (
Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
192190, 191syl 16 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
193189, 192eleqtrrd 2520 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
19430gsumwmhm 15528 . . . . . . . 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
) ) ) )
195175, 193, 194syl2anc 661 . . . . . . 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
) ) ) )
19612, 13efger 16220 . . . . . . . . 9  |-  .~  Er  W
197196a1i 11 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  .~  Er  W )
198 fvex 5706 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
19912, 198eqeltri 2513 . . . . . . . . 9  |-  W  e. 
_V
200199a1i 11 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  W  e.  _V )
201 eqid 2443 . . . . . . . 8  |-  ( w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  )
202197, 200, 201divsfval 14490 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) `  t )  =  [ t ]  .~  )
203187, 195, 2023eqtr3d 2483 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  [ t ]  .~  )
204203fveq2d 5700 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) )  =  ( K `  [
t ]  .~  )
)
205184, 204eqtr2d 2476 . . . 4  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  [ t ]  .~  )  =  ( E `  [ t ]  .~  ) )
20644, 47, 205ectocld 7172 . . 3  |-  ( (
ph  /\  a  e.  ( W /.  .~  )
)  ->  ( K `  a )  =  ( E `  a ) )
20743, 206syldan 470 . 2  |-  ( (
ph  /\  a  e.  X )  ->  ( K `  a )  =  ( E `  a ) )
2086, 19, 207eqfnfvd 5805 1  |-  ( ph  ->  K  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   ifcif 3796   {cpr 3884   <.cop 3888    e. cmpt 4355    _I cid 4636   Oncon0 4724    X. cxp 4843   ran crn 4846    o. ccom 4849    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1oc1o 6918   2oc2o 6919    Er wer 7103   [cec 7104   /.cqs 7105   0cc0 9287  ..^cfzo 11553   #chash 12108  Word cword 12226   <"cs1 12229   Basecbs 14179    gsumg cgsu 14384    /.s cqus 14448   Grpcgrp 15415   invgcminusg 15416   MndHom cmhm 15467  freeMndcfrmd 15530  varFMndcvrmd 15531    GrpHom cghm 15749   ~FG cefg 16208  freeGrpcfrgp 16209  varFGrpcvrgp 16210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-ot 3891  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-ec 7108  df-qs 7112  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-word 12234  df-concat 12236  df-s1 12237  df-substr 12238  df-splice 12239  df-reverse 12240  df-s2 12480  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-0g 14385  df-gsum 14386  df-imas 14451  df-divs 14452  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-frmd 15532  df-vrmd 15533  df-grp 15550  df-minusg 15551  df-ghm 15750  df-efg 16211  df-frgp 16212  df-vrgp 16213
This theorem is referenced by:  frgpup3  16280
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