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Theorem frgpup3lem 16664
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 16140 . . 3  |-  ( K  e.  ( G  GrpHom  H )  ->  K : X
--> B )
5 ffn 5717 . . 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 16662 . . 3  |-  ( ph  ->  E  e.  ( G 
GrpHom  H ) )
172, 3ghmf 16140 . . 3  |-  ( E  e.  ( G  GrpHom  H )  ->  E : X
--> B )
18 ffn 5717 . . 3  |-  ( E : X --> B  ->  E  Fn  X )
1916, 17, 183syl 20 . 2  |-  ( ph  ->  E  Fn  X )
20 eqid 2441 . . . . . . . . 9  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
2114, 20, 13frgpval 16645 . . . . . . . 8  |-  ( I  e.  V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
2210, 21syl 16 . . . . . . 7  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
23 2on 7136 . . . . . . . . . . 11  |-  2o  e.  On
24 xpexg 6583 . . . . . . . . . . 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 12531 . . . . . . . . . 10  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
27 fvi 5911 . . . . . . . . . 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 2494 . . . . . . . 8  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
30 eqid 2441 . . . . . . . . . 10  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
3120, 30frmdbas 15889 . . . . . . . . 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 2485 . . . . . . 7  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
34 fvex 5862 . . . . . . . . 9  |-  ( ~FG  `  I
)  e.  _V
3513, 34eqeltri 2525 . . . . . . . 8  |-  .~  e.  _V
3635a1i 11 . . . . . . 7  |-  ( ph  ->  .~  e.  _V )
37 fvex 5862 . . . . . . . 8  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
3837a1i 11 . . . . . . 7  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
3922, 33, 36, 38qusbas 14814 . . . . . 6  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
4039, 2syl6reqr 2501 . . . . 5  |-  ( ph  ->  X  =  ( W /.  .~  ) )
41 eqimss 3538 . . . . 5  |-  ( X  =  ( W /.  .~  )  ->  X  C_  ( W /.  .~  ) )
4240, 41syl 16 . . . 4  |-  ( ph  ->  X  C_  ( W /.  .~  ) )
4342sselda 3486 . . 3  |-  ( (
ph  /\  a  e.  X )  ->  a  e.  ( W /.  .~  ) )
44 eqid 2441 . . . 4  |-  ( W /.  .~  )  =  ( W /.  .~  )
45 fveq2 5852 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( K `  [ t ]  .~  )  =  ( K `  a ) )
46 fveq2 5852 . . . . 5  |-  ( [ t ]  .~  =  a  ->  ( E `  [ t ]  .~  )  =  ( E `  a ) )
4745, 46eqeq12d 2463 . . . 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 2531 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  t  e. Word  ( I  X.  2o ) )
51 wrdf 12527 . . . . . . . . . . . . 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 6012 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( t `  n )  e.  ( I  X.  2o ) )
54 elxp2 5003 . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( F `  y )  =  ( F `  i ) )
5756fveq2d 5856 . . . . . . . . . . . . . . . . 17  |-  ( y  =  i  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  i )
) )
5856, 57ifeq12d 3942 . . . . . . . . . . . . . . . 16  |-  ( y  =  i  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  if ( z  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
59 eqeq1 2445 . . . . . . . . . . . . . . . . 17  |-  ( z  =  j  ->  (
z  =  (/)  <->  j  =  (/) ) )
6059ifbid 3944 . . . . . . . . . . . . . . . 16  |-  ( z  =  j  ->  if ( z  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  if ( j  =  (/) ,  ( F `
 i ) ,  ( N `  ( F `  i )
) ) )
61 fvex 5862 . . . . . . . . . . . . . . . . 17  |-  ( F `
 i )  e. 
_V
62 fvex 5862 . . . . . . . . . . . . . . . . 17  |-  ( N `
 ( F `  i ) )  e. 
_V
6361, 62ifex 3991 . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  if ( j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) ) )
66 elpri 4030 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  { (/) ,  1o }  ->  ( j  =  (/)  \/  j  =  1o ) )
67 df2o3 7141 . . . . . . . . . . . . . . . . 17  |-  2o  =  { (/) ,  1o }
6866, 67eleq2s 2549 . . . . . . . . . . . . . . . 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 5854 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  (
( K  o.  U
) `  i )  =  ( F `  i ) )
72 frgpup.u . . . . . . . . . . . . . . . . . . . . . . 23  |-  U  =  (varFGrp `  I )
7313, 72, 14, 2vrgpf 16655 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( I  e.  V  ->  U : I --> X )
7410, 73syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  U : I --> X )
75 fvco3 5931 . . . . . . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . . . . 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 3928 . . . . . . . . . . . . . . . . . . 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 4205 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <. i ,  j >.  =  <. i ,  (/) >. )
8382s1eqd 12587 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  <" <. i ,  j >. ">  =  <" <. i ,  (/) >. "> )
8483eceq1d 7346 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  (/) >. "> ]  .~  )
8513, 72vrgpval 16654 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( U `  i
)  =  [ <"
<. i ,  (/) >. "> ]  .~  )
8610, 85sylan 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8786adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( U `  i )  =  [ <" <. i ,  (/) >. "> ]  .~  )
8884, 87eqtr4d 2485 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( U `  i ) )
8988fveq2d 5856 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( U `
 i ) ) )
9078, 80, 893eqtr4d 2492 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  (/) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
9177fveq2d 5856 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( N `  ( K `  ( U `  i ) ) ) )
921adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  K  e.  ( G  GrpHom  H ) )
9374ffvelrnda 6012 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  ( U `  i )  e.  X )
94 eqid 2441 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( invg `  G )  =  ( invg `  G )
952, 94, 7ghminv 16143 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( K  e.  ( G 
GrpHom  H )  /\  ( U `  i )  e.  X )  ->  ( K `  ( ( invg `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9692, 93, 95syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  i  e.  I )  ->  ( K `  ( ( invg `  G ) `
 ( U `  i ) ) )  =  ( N `  ( K `  ( U `
 i ) ) ) )
9791, 96eqtr4d 2485 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  i  e.  I )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( invg `  G ) `  ( U `  i )
) ) )
9897adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( N `  ( F `  i ) )  =  ( K `  (
( invg `  G ) `  ( U `  i )
) ) )
99 1n0 7143 . . . . . . . . . . . . . . . . . . . 20  |-  1o  =/=  (/)
100 simpr 461 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =  1o )
101100neeq1d 2718 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
j  =/=  (/)  <->  1o  =/=  (/) ) )
10299, 101mpbiri 233 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  j  =/=  (/) )
103 ifnefalse 3934 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =/=  (/)  ->  if (
j  =  (/) ,  ( F `  i ) ,  ( N `  ( F `  i ) ) )  =  ( N `  ( F `
 i ) ) )
104102, 103syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( N `  ( F `  i ) ) )
105100opeq2d 4205 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <. i ,  j >.  =  <. i ,  1o >. )
106105s1eqd 12587 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  <" <. i ,  j >. ">  =  <" <. i ,  1o >. "> )
107106eceq1d 7346 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  [ <" <. i ,  1o >. "> ]  .~  )
10813, 72, 14, 94vrgpinv 16656 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  e.  V  /\  i  e.  I )  ->  ( ( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
10910, 108sylan 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  i  e.  I )  ->  (
( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
110109adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  (
( invg `  G ) `  ( U `  i )
)  =  [ <"
<. i ,  1o >. "> ]  .~  )
111107, 110eqtr4d 2485 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  [ <"
<. i ,  j >. "> ]  .~  =  ( ( invg `  G ) `  ( U `  i )
) )
112111fveq2d 5856 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  ( K `  [ <" <. i ,  j >. "> ]  .~  )  =  ( K `  ( ( invg `  G
) `  ( U `  i ) ) ) )
11398, 104, 1123eqtr4d 2492 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  i  e.  I )  /\  j  =  1o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11490, 113jaodan 783 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  i  e.  I )  /\  (
j  =  (/)  \/  j  =  1o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11568, 114sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  i  e.  I )  /\  j  e.  2o )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
116115anasss 647 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  ->  if ( j  =  (/) ,  ( F `  i
) ,  ( N `
 ( F `  i ) ) )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
11765, 116eqtrd 2482 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( i T j )  =  ( K `
 [ <" <. i ,  j >. "> ]  .~  ) )
118 fveq2 5852 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( T `  <. i ,  j >. )
)
119 df-ov 6280 . . . . . . . . . . . . . . 15  |-  ( i T j )  =  ( T `  <. i ,  j >. )
120118, 119syl6eqr 2500 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( T `  ( t `  n
) )  =  ( i T j ) )
121 s1eq 12586 . . . . . . . . . . . . . . . 16  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  <" ( t `
 n ) ">  =  <" <. i ,  j >. "> )
122121eceq1d 7346 . . . . . . . . . . . . . . 15  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  [ <" (
t `  n ) "> ]  .~  =  [ <" <. i ,  j >. "> ]  .~  )
123122fveq2d 5856 . . . . . . . . . . . . . 14  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( K `  [ <" ( t `
 n ) "> ]  .~  )  =  ( K `  [ <" <. i ,  j >. "> ]  .~  ) )
124120, 123eqeq12d 2463 . . . . . . . . . . . . 13  |-  ( ( t `  n )  =  <. i ,  j
>.  ->  ( ( T `
 ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  )  <->  ( i T j )  =  ( K `  [ <" <. i ,  j
>. "> ]  .~  ) ) )
125117, 124syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  e.  I  /\  j  e.  2o ) )  -> 
( ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
126125rexlimdvva 2940 . . . . . . . . . . 11  |-  ( ph  ->  ( E. i  e.  I  E. j  e.  2o  ( t `  n )  =  <. i ,  j >.  ->  ( T `  ( t `  n ) )  =  ( K `  [ <" ( t `  n ) "> ]  .~  ) ) )
127126ad2antrr 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 ) "> ]  .~  ) ) )
12855, 127mpd 15 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  ( T `  ( t `  n
) )  =  ( K `  [ <" ( t `  n
) "> ]  .~  ) )
129128mpteq2dva 4519 . . . . . . . 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 16657 . . . . . . . . . 10  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
131130adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  T : ( I  X.  2o ) --> B )
132 fcompt 6048 . . . . . . . . 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 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( T `  ( t `
 n ) ) ) )
13453s1cld 12589 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  <" ( t `
 n ) ">  e. Word  ( I  X.  2o ) )
13529ad2antrr 725 . . . . . . . . . . 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 16648 . . . . . . . . . 10  |-  ( <" ( t `  n ) ">  e.  W  ->  [ <" ( t `  n
) "> ]  .~  e.  X )
138136, 137syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  W )  /\  n  e.  ( 0..^ ( # `  t ) ) )  ->  [ <" (
t `  n ) "> ]  .~  e.  X )
13952feqmptd 5907 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  t  =  ( n  e.  ( 0..^ ( # `  t ) )  |->  ( t `  n ) ) )
14010adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  W )  ->  I  e.  V )
141140, 23, 24sylancl 662 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  (
I  X.  2o )  e.  _V )
142 eqid 2441 . . . . . . . . . . . . 13  |-  (varFMnd `  (
I  X.  2o ) )  =  (varFMnd `  (
I  X.  2o ) )
143142vrmdfval 15893 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
144141, 143syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) )  =  ( w  e.  ( I  X.  2o )  |->  <" w "> ) )
145 s1eq 12586 . . . . . . . . . . 11  |-  ( w  =  ( t `  n )  ->  <" w ">  =  <" (
t `  n ) "> )
14653, 139, 144, 145fmptco 6045 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  <" ( t `
 n ) "> ) )
147 eqidd 2442 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  ) )
148 eceq1 7345 . . . . . . . . . 10  |-  ( w  =  <" ( t `
 n ) ">  ->  [ w ]  .~  =  [ <" ( t `  n
) "> ]  .~  )
149136, 146, 147, 148fmptco 6045 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  =  ( n  e.  ( 0..^ (
# `  t )
)  |->  [ <" (
t `  n ) "> ]  .~  )
)
1501adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G  GrpHom  H ) )
151150, 4syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  K : X --> B )
152151feqmptd 5907 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  K  =  ( w  e.  X  |->  ( K `  w ) ) )
153 fveq2 5852 . . . . . . . . 9  |-  ( w  =  [ <" (
t `  n ) "> ]  .~  ->  ( K `  w )  =  ( K `  [ <" ( t `
 n ) "> ]  .~  )
)
154138, 149, 152, 153fmptco 6045 . . . . . . . 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 2492 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  ( T  o.  t )  =  ( K  o.  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) )
156155oveq2d 6293 . . . . . 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 16661 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( H  gsumg  ( T  o.  t
) ) )
158 ghmmhm 16146 . . . . . . . 8  |-  ( K  e.  ( G  GrpHom  H )  ->  K  e.  ( G MndHom  H ) )
159150, 158syl 16 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  K  e.  ( G MndHom  H ) )
160142vrmdf 15895 . . . . . . . . . . 11  |-  ( ( I  X.  2o )  e.  _V  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
161141, 160syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) -->Word  ( I  X.  2o ) )
16249feq3d 5705 . . . . . . . . . 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 232 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (varFMnd `  (
I  X.  2o ) ) : ( I  X.  2o ) --> W )
164 wrdco 12771 . . . . . . . . 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 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word  W )
16633adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
167166mpteq1d 4514 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
)
168 eqid 2441 . . . . . . . . . . . . 13  |-  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )  =  ( w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  |->  [ w ]  .~  )
16920, 30, 14, 13, 168frgpmhm 16652 . . . . . . . . . . . 12  |-  ( I  e.  V  ->  (
w  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) 
|->  [ w ]  .~  )  e.  ( (freeMnd `  ( I  X.  2o ) ) MndHom  G ) )
170140, 169syl 16 . . . . . . . . . . 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 15840 . . . . . . . . . 10  |-  ( ( w  e.  W  |->  [ w ]  .~  )  e.  ( (freeMnd `  (
I  X.  2o ) ) MndHom  G )  -> 
( w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
173171, 172syl 16 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : ( Base `  (freeMnd `  ( I  X.  2o ) ) ) --> X )
174166feq2d 5704 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) : W --> X  <->  ( w  e.  W  |->  [ w ]  .~  ) : (
Base `  (freeMnd `  (
I  X.  2o ) ) ) --> X ) )
175173, 174mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
w  e.  W  |->  [ w ]  .~  ) : W --> X )
176 wrdco 12771 . . . . . . . 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 661 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) )  e. Word  X )
1782gsumwmhm 15882 . . . . . . 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 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 ) ) ) ) )
180156, 157, 1793eqtr4d 2492 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( E `  [ t ]  .~  )  =  ( K `  ( G 
gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) ) ) )
18120, 142frmdgsum 15899 . . . . . . . . 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 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  (
(freeMnd `  ( I  X.  2o ) )  gsumg  ( (varFMnd `  ( I  X.  2o ) )  o.  t
) )  =  t )
183182fveq2d 5856 . . . . . . 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 12771 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  W )  ->  (
(varFMnd `  ( I  X.  2o ) )  o.  t
)  e. Word Word  ( I  X.  2o ) )
18632adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  W )  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
187 wrdeq 12538 . . . . . . . . . 10  |-  ( (
Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o )  -> Word  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word Word  ( I  X.  2o ) )
188186, 187syl 16 . . . . . . . . 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 15882 . . . . . . . 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 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
) ) ) )
19212, 13efger 16605 . . . . . . . . 9  |-  .~  Er  W
193192a1i 11 . . . . . . . 8  |-  ( (
ph  /\  t  e.  W )  ->  .~  Er  W )
194 fvex 5862 . . . . . . . . . 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 2441 . . . . . . . 8  |-  ( w  e.  W  |->  [ w ]  .~  )  =  ( w  e.  W  |->  [ w ]  .~  )
198193, 196, 197divsfval 14816 . . . . . . 7  |-  ( (
ph  /\  t  e.  W )  ->  (
( w  e.  W  |->  [ w ]  .~  ) `  t )  =  [ t ]  .~  )
199183, 191, 1983eqtr3d 2490 . . . . . 6  |-  ( (
ph  /\  t  e.  W )  ->  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  (
(varFMnd `  ( I  X.  2o ) )  o.  t
) ) )  =  [ t ]  .~  )
200199fveq2d 5856 . . . . 5  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  ( G  gsumg  ( ( w  e.  W  |->  [ w ]  .~  )  o.  ( (varFMnd `  (
I  X.  2o ) )  o.  t ) ) ) )  =  ( K `  [
t ]  .~  )
)
201180, 200eqtr2d 2483 . . . 4  |-  ( (
ph  /\  t  e.  W )  ->  ( K `  [ t ]  .~  )  =  ( E `  [ t ]  .~  ) )
20244, 47, 201ectocld 7376 . . 3  |-  ( (
ph  /\  a  e.  ( W /.  .~  )
)  ->  ( K `  a )  =  ( E `  a ) )
20343, 202syldan 470 . 2  |-  ( (
ph  /\  a  e.  X )  ->  ( K `  a )  =  ( E `  a ) )
2046, 19, 203eqfnfvd 5965 1  |-  ( ph  ->  K  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792   _Vcvv 3093    C_ wss 3458   (/)c0 3767   ifcif 3922   {cpr 4012   <.cop 4016    |-> cmpt 4491    _I cid 4776   Oncon0 4864    X. cxp 4983   ran crn 4986    o. ccom 4989    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   1oc1o 7121   2oc2o 7122    Er wer 7306   [cec 7307   /.cqs 7308   0cc0 9490  ..^cfzo 11798   #chash 12379  Word cword 12508   <"cs1 12511   Basecbs 14504    gsumg cgsu 14710    /.s cqus 14774   MndHom cmhm 15833  freeMndcfrmd 15884  varFMndcvrmd 15885   Grpcgrp 15922   invgcminusg 15923    GrpHom cghm 16133   ~FG cefg 16593  freeGrpcfrgp 16594  varFGrpcvrgp 16595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-ot 4019  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-ec 7311  df-qs 7315  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-word 12516  df-concat 12518  df-s1 12519  df-substr 12520  df-splice 12521  df-reverse 12522  df-s2 12787  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-0g 14711  df-gsum 14712  df-imas 14777  df-qus 14778  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-submnd 15836  df-frmd 15886  df-vrmd 15887  df-grp 15926  df-minusg 15927  df-ghm 16134  df-efg 16596  df-frgp 16597  df-vrgp 16598
This theorem is referenced by:  frgpup3  16665
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