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Theorem frgpup3lem 15086
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  =  ( inv g `  H )
frgpup.t  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
frgpup.h  |-  ( ph  ->  H  e.  Grp )
frgpup.i  |-  ( ph  ->  I  e.  V )
frgpup.a  |-  ( ph  ->  F : I --> B )
frgpup.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpup.r  |-  .~  =  ( ~FG  `  I )
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 14687 . . 3  |-  ( K  e.  ( G  GrpHom  H )  ->  K : X
--> B )
5 ffn 5389 . . 3  |-  ( K : X --> B  ->  K  Fn  X )
61, 4, 53syl 18 . 2  |-  ( ph  ->  K  Fn  X )
7 frgpup.n . . . 4  |-  N  =  ( inv g `  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 15084 . . 3  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
172, 3ghmf 14687 . . 3  |-  ( E  e.  ( G  GrpHom  H )  ->  E : X
--> B )
18 ffn 5389 . . 3  |-  ( E : X --> B  ->  E  Fn  X )
1916, 17, 183syl 18 . 2  |-  ( ph  ->  E  Fn  X )
20 eqid 2283 . . . . . . . . 9  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
2114, 20, 13frgpval 15067 . . . . . . . 8  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
2210, 21syl 15 . . . . . . 7  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
23 2on 6487 . . . . . . . . . . 11  |-  2o  e.  On
24 xpexg 4800 . . . . . . . . . . 11  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
2510, 23, 24sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
26 wrdexg 11425 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
27 fvi 5579 . . . . . . . . . 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 2327 . . . . . . . 8  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
30 eqid 2283 . . . . . . . . . 10  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
3120, 30frmdbas 14474 . . . . . . . . 9  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
3225, 31syl 15 . . . . . . . 8  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
3329, 32eqtr4d 2318 . . . . . . 7  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
34 fvex 5539 . . . . . . . . 9  |-  ( ~FG  `  I
)  e.  _V
3513, 34eqeltri 2353 . . . . . . . 8  |-  .~  e.  _V
3635a1i 10 . . . . . . 7  |-  ( ph  ->  .~  e.  _V )
37 fvex 5539 . . . . . . . 8  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3837a1i 10 . . . . . . 7  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3922, 33, 36, 38divsbas 13447 . . . . . 6  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
4039, 2syl6reqr 2334 . . . . 5  |-  ( ph  ->  X  =  ( W /.  .~  ) )
41 eqimss 3230 . . . . 5  |-  ( X  =  ( W /.  .~  )  ->  X  C_  ( W /.  .~  ) )
4240, 41syl 15 . . . 4  |-  ( ph  ->  X  C_  ( W /.  .~  ) )
4342sselda 3180 . . 3  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
44 eqid 2283 . . . 4  |-  ( W /.  .~  )  =  ( W /.  .~  )
45 fveq2 5525 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( K `  [ t ]  .~  )  =  ( K `  a ) )
46 fveq2 5525 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
4745, 46eqeq12d 2297 . . . 4  |-  ( [ t ]  .~  =  a  ->  ( ( K `
 [ t ]  .~  )  =  ( E `  [ t ]  .~  )  <->  ( K `  a )  =  ( E `  a ) ) )
48 simpr 447 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  W )  ->  t  e.  W )
4929adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  W )  ->  W  = Word  ( I  X.  2o ) )
5048, 49eleqtrd 2359 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  t  e. Word  ( I  X.  2o ) )
51 wrdf 11419 . . . . . . . . . . . . 13  |-  ( t  e. Word  ( I  X.  2o )  ->  t : ( 0..^ ( # `  t ) ) --> ( I  X.  2o ) )
5250, 51syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  t : ( 0..^ (
# `  t )
) --> ( I  X.  2o ) )
53 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( t : ( 0..^ ( # `  t
) ) --> ( I  X.  2o )  /\  n  e.  ( 0..^ ( # `  t
) ) )  -> 
( t `  n
)  e.  ( I  X.  2o ) )
5452, 53sylan 457 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( t `  n )  e.  ( I  X.  2o ) )
55 elxp2 4707 . . . . . . . . . . 11  |-  ( ( t `  n )  e.  ( I  X.  2o )  <->  E. i  e.  I  E. j  e.  2o  ( t `  n
)  =  <. i ,  j >. )
5654, 55sylib 188 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  E. i  e.  I  E. j  e.  2o  ( t `  n
)  =  <. i ,  j >. )
57 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( F `  y )  =  ( F `  i ) )
5857fveq2d 5529 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  i )
) )
5957, 58ifeq12d 3581 . . . . . . . . . . . . . . . 16  |-  ( y  =  i  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  if ( z  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
60 eqeq1 2289 . . . . . . . . . . . . . . . . 17  |-  ( z  =  j  ->  (
z  =  (/)  <->  j  =  (/) ) )
6160ifbid 3583 . . . . . . . . . . . . . . . 16  |-  ( z  =  j  ->  if ( z  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  if ( j  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
62 fvex 5539 . . . . . . . . . . . . . . . . 17  |-  ( F `
 i )  e. 
_V
63 fvex 5539 . . . . . . . . . . . . . . . . 17  |-  ( N `
 ( F `  i ) )  e. 
_V
6462, 63ifex 3623 . . . . . . . . . . . . . . . 16  |-  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  e.  _V
6559, 61, 8, 64ovmpt2 5983 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  I  /\  j  e.  2o )  ->  ( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
6665adantl 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
67 elpri 3660 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  { (/) ,  1o }  ->  ( j  =  (/)  \/  j  =  1o ) )
68 df2o3 6492 . . . . . . . . . . . . . . . . 17  |-  2o  =  { (/) ,  1o }
6967, 68eleq2s 2375 . . . . . . . . . . . . . . . 16  |-  ( j  e.  2o  ->  (
j  =  (/)  \/  j  =  1o ) )
70 frgpup3.e . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( K  o.  U
)  =  F )
7170adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( K  o.  U )  =  F )
7271fveq1d 5527 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( F `  i ) )
73 frgpup.u . . . . . . . . . . . . . . . . . . . . . . 23  |-  U  =  (varFGrp `  I )
7413, 73, 14, 2vrgpf 15077 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( I  e.  V  ->  U : I --> X )
7510, 74syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  U : I --> X )
76 fvco3 5596 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U : I --> X  /\  i  e.  I )  ->  ( ( K  o.  U ) `  i
)  =  ( K `
 ( U `  i ) ) )
7775, 76sylan 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( K `  ( U `  i ) ) )
7872, 77eqtr3d 2317 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( F `  i )  =  ( K `  ( U `  i ) ) )
7978adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( F `  i )  =  ( K `  ( U `  i ) ) )
80 iftrue 3571 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  (/)  ->  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( F `  i ) )
8180adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( F `  i ) )
82 simpr 447 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  j  =  (/) )
8382opeq2d 3803 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <. i ,  j >.  =  <. i ,  (/) >. )
8483s1eqd 11440 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <" <. i ,  j >. ">  =  <" <. i ,  (/) >. "> )
85 eceq1 6696 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" <. i ,  j
>. ">  =  <"
<. i ,  (/) >. ">  ->  [ <" <. i ,  j >. "> ]  .~  =  [ <"
<. i ,  (/) >. "> ]  .~  )
8684, 85syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  (/) >. "> ]  .~  )
8713, 73vrgpval 15076 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( U `  i
)  =  [ <"
<. i ,  (/) >. "> ]  .~  )
8810, 87sylan 457 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8988adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
9086, 89eqtr4d 2318 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( U `  i ) )
9190fveq2d 5529 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( U `
 i ) ) )
9279, 81, 913eqtr4d 2325 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
9378fveq2d 5529 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( N `  ( K `  ( U `  i ) ) ) )
941adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  K  e.  ( G  GrpHom  H ) )
95 ffvelrn 5663 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( U : I --> X  /\  i  e.  I )  ->  ( U `  i
)  e.  X )
9675, 95sylan 457 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  e.  X )
97 eqid 2283 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( inv g `  G )  =  ( inv g `  G )
982, 97, 7ghminv 14690 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( K  e.  ( G 
GrpHom  H )  /\  ( U `  i )  e.  X )  ->  ( K `  ( ( inv g `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9994, 96, 98syl2anc 642 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( K `  ( ( inv g `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
10093, 99eqtr4d 2318 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( inv g `  G ) `  ( U `  i )
) ) )
101100adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( inv g `  G ) `  ( U `  i )
) ) )
102 1n0 6494 . . . . . . . . . . . . . . . . . . . 20  |-  1o  =/=  (/)
103 simpr 447 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =  1o )
104103neeq1d 2459 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
j  =/=  (/)  <->  1o  =/=  (/) ) )
105102, 104mpbiri 224 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =/=  (/) )
106 ifnefalse 3573 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =/=  (/)  ->  if (
j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( N `  ( F `
 i ) ) )
107105, 106syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( N `  ( F `  i ) ) )
108103opeq2d 3803 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <. i ,  j >.  =  <. i ,  1o >. )
109108s1eqd 11440 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <" <. i ,  j >. ">  =  <" <. i ,  1o >. "> )
110 eceq1 6696 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <" <. i ,  j
>. ">  =  <"
<. i ,  1o >. ">  ->  [ <" <. i ,  j >. "> ]  .~  =  [ <"
<. i ,  1o >. "> ]  .~  )
111109, 110syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  1o >. "> ]  .~  )
11213, 73, 14, 97vrgpinv 15078 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( ( inv g `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
11310, 112sylan 457 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  (
( inv g `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
114113adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
( inv g `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
115111, 114eqtr4d 2318 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( ( inv g `  G ) `  ( U `  i )
) )
116115fveq2d 5529 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( ( inv g `  G
) `  ( U `  i ) ) ) )
117101, 107, 1163eqtr4d 2325 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11892, 117jaodan 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  I )  /\  (
j  =  (/)  \/  j  =  1o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11969, 118sylan2 460 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  I )  /\  j  e.  2o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
120119anasss 628 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
12166, 120eqtrd 2315 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  ( K `
 [ <" <. i ,  j >. "> ]  .~  ) )
122 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( T `  <. i ,  j >. )
)
123 df-ov 5861 . . . . . . . . . . . . . . 15  |-  ( i T j )  =  ( T `  <. i ,  j >. )
124122, 123syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( i T j ) )
125 s1eq 11439 . . . . . . . . . . . . . . . 16  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  <" ( t `
 n ) ">  =  <" <. i ,  j >. "> )
126 eceq1 6696 . . . . . . . . . . . . . . . 16  |-  ( <" ( t `  n ) ">  =  <" <. i ,  j >. ">  ->  [ <" (
t `  n ) "> ]  .~  =  [ <" <. i ,  j >. "> ]  .~  )
127125, 126syl 15 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  [ <" (
t `  n ) "> ]  .~  =  [ <" <. i ,  j >. "> ]  .~  )
128127fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( K `  [ <" ( t `
 n ) "> ]  .~  )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
129124, 128eqeq12d 2297 . . . . . . . . . . . . 13  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( ( T `
 ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  )  <->  ( i T j )  =  ( K `  [ <" <. i ,  j
>. "> ]  .~  ) ) )
130121, 129syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
131130rexlimdvva 2674 . . . . . . . . . . 11  |-  ( ph  ->  ( E. i  e.  I  E. j  e.  2o  ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
132131ad2antrr 706 . . . . . . . . . 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 ) "> ]  .~  ) ) )
13356, 132mpd 14 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( T `  ( t `  n
) )  =  ( K `  [ <" ( t `  n
) "> ]  .~  ) )
134133mpteq2dva 4106 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
n  e.  ( 0..^ ( # `  t
) )  |->  ( T `
 ( t `  n ) ) )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( K `  [ <" ( t `  n
) "> ]  .~  ) ) )
1353, 7, 8, 9, 10, 11frgpuptf 15079 . . . . . . . . . 10  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
136135adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  T : ( I  X.  2o ) --> B )
137 fcompt 5694 . . . . . . . . 9  |-  ( ( T : ( I  X.  2o ) --> B  /\  t : ( 0..^ ( # `  t
) ) --> ( I  X.  2o ) )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t
) )  |->  ( T `
 ( t `  n ) ) ) )
138136, 52, 137syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( T `  ( t `
 n ) ) ) )
13954s1cld 11442 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e. Word  ( I  X.  2o ) )
14029ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  W  = Word  (
I  X.  2o ) )
141139, 140eleqtrrd 2360 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e.  W )
14214, 13, 12, 2frgpeccl 15070 . . . . . . . . . 10  |-  ( <" ( t `  n ) ">  e.  W  ->  [ <" ( t `  n
) "> ]  .~  e.  X )
143141, 142syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  [ <" (
t `  n ) "> ]  .~  e.  X )
14452feqmptd 5575 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  t  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( t `  n ) ) )
14510adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  I  e.  V )
146145, 23, 24sylancl 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  (
I  X.  2o )  e.  _V )
147 eqid 2283 . . . . . . . . . . . . 13  |-  (varFMnd `  (
I  X.  2o ) )  =  (varFMnd `  (
I  X.  2o ) )
148147vrmdfval 14478 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
149146, 148syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
150 s1eq 11439 . . . . . . . . . . 11  |-  ( w  =  ( t `  n )  ->  <" w ">  =  <" (
t `  n ) "> )
15154, 144, 149, 150fmptco 5691 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  <" ( t `
 n ) "> ) )
152 eqidd 2284 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  ) )
153 eceq1 6696 . . . . . . . . . 10  |-  ( w  =  <" ( t `
 n ) ">  ->  [ w ]  .~  =  [ <" ( t `  n
) "> ]  .~  )
154141, 151, 152, 153fmptco 5691 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  [ <" (
t `  n ) "> ]  .~  )
)
1551adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G  GrpHom  H ) )
156155, 4syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  K : X --> B )
157156feqmptd 5575 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  K  =  ( w  e.  X  |->  ( K `  w ) ) )
158 fveq2 5525 . . . . . . . . 9  |-  ( w  =  [ <" (
t `  n ) "> ]  .~  ->  ( K `  w )  =  ( K `  [ <" ( t `
 n ) "> ]  .~  )
)
159143, 154, 157, 158fmptco 5691 . . . . . . . 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 ) "> ]  .~  )
) )
160134, 138, 1593eqtr4d 2325 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( K  o.  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )
161160oveq2d 5874 . . . . . 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 ) ) ) ) )
1623, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15frgpupval 15083 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( H  gsumg  ( T  o.  t
) ) )
163 ghmmhm 14693 . . . . . . . 8  |-  ( K  e.  ( G  GrpHom  H )  ->  K  e.  ( G MndHom  H ) )
164155, 163syl 15 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G MndHom  H ) )
165147vrmdf 14480 . . . . . . . . . . 11  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
166146, 165syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
167 feq3 5377 . . . . . . . . . . 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 ) ) )
16849, 167syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W  <->  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) ) )
169166, 168mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) --> W )
170 wrdco 11486 . . . . . . . . 9  |-  ( ( t  e. Word  ( I  X.  2o )  /\  (varFMnd `  ( I  X.  2o ) ) : ( I  X.  2o ) --> W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
17150, 169, 170syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
17233adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
173 mpteq1 4100 . . . . . . . . . . . 12  |-  ( W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  -> 
( w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
)
174172, 173syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
)
175 eqid 2283 . . . . . . . . . . . . 13  |-  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
17620, 30, 14, 13, 175frgpmhm 15074 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
177145, 176syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
178174, 177eqeltrd 2357 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G ) )
17930, 2mhmf 14420 . . . . . . . . . 10  |-  ( ( w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G )  -> 
( w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
180178, 179syl 15 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
181172feq2d 5380 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) : W --> X  <->  ( w  e.  W  |->  [ w ]  .~  ) : (
Base `  (freeMnd `  (
I  X.  2o ) ) ) --> X ) )
182180, 181mpbird 223 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : W --> X )
183 wrdco 11486 . . . . . . . 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
)
184171, 182, 183syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  e. Word  X )
1852gsumwmhm 14467 . . . . . . 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
) ) ) ) )
186164, 184, 185syl2anc 642 . . . . . 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 ) ) ) ) )
187161, 162, 1863eqtr4d 2325 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( K `  ( G 
gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) ) )
18820, 147frmdgsum 14484 . . . . . . . . 9  |-  ( ( ( I  X.  2o )  e.  _V  /\  t  e. Word  ( I  X.  2o ) )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
189146, 50, 188syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
190189fveq2d 5529 . . . . . . 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 ) )
191 wrdco 11486 . . . . . . . . . 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 ) )
19250, 166, 191syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word Word  ( I  X.  2o ) )
19332adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
194 wrdeq 11424 . . . . . . . . . 10  |-  ( (
Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
195193, 194syl 15 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
196192, 195eleqtrrd 2360 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
19730gsumwmhm 14467 . . . . . . . 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
) ) ) )
198178, 196, 197syl2anc 642 . . . . . . 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
) ) ) )
19912, 13efger 15027 . . . . . . . . 9  |-  .~  Er  W
200199a1i 10 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  .~  Er  W )
201 fvex 5539 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
20212, 201eqeltri 2353 . . . . . . . . 9  |-  W  e. 
_V
203202a1i 10 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  W  e.  _V )
204 eqid 2283 . . . . . . . 8  |-  ( w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  )
205200, 203, 204divsfval 13449 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) `  t )  =  [ t ]  .~  )
206190, 198, 2053eqtr3d 2323 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  [ t ]  .~  )
207206fveq2d 5529 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) )  =  ( K `  [
t ]  .~  )
)
208187, 207eqtr2d 2316 . . . 4  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  [ t ]  .~  )  =  ( E `  [ t ]  .~  ) )
20944, 47, 208ectocld 6726 . . 3  |-  ( (
ph  /\  a  e.  ( W /.  .~  )
)  ->  ( K `  a )  =  ( E `  a ) )
21043, 209syldan 456 . 2  |-  ( (
ph  /\  a  e.  X )  ->  ( K `  a )  =  ( E `  a ) )
2116, 19, 210eqfnfvd 5625 1  |-  ( ph  ->  K  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ifcif 3565   {cpr 3641   <.cop 3643    e. cmpt 4077    _I cid 4304   Oncon0 4392    X. cxp 4687   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   [cec 6658   /.cqs 6659   0cc0 8737  ..^cfzo 10870   #chash 11337  Word cword 11403   <"cs1 11405   Basecbs 13148    gsumg cgsu 13401    /.s cqus 13408   Grpcgrp 14362   inv gcminusg 14363   MndHom cmhm 14413  freeMndcfrmd 14469  varFMndcvrmd 14470    GrpHom cghm 14680   ~FG cefg 15015  freeGrpcfrgp 15016  varFGrpcvrgp 15017
This theorem is referenced by:  frgpup3  15087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-frmd 14471  df-vrmd 14472  df-grp 14489  df-minusg 14490  df-ghm 14681  df-efg 15018  df-frgp 15019  df-vrgp 15020
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