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Theorem frgpup2 16609
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 )
frgpup.y  |-  ( ph  ->  A  e.  I )
Assertion
Ref Expression
frgpup2  |-  ( ph  ->  ( E `  ( U `  A )
)  =  ( F `
 A ) )
Distinct variable groups:    y, g,
z, A    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)    N( g)    V( y, z, g)    W( y, z)    X( y, z, g)

Proof of Theorem frgpup2
StepHypRef Expression
1 frgpup.i . . . 4  |-  ( ph  ->  I  e.  V )
2 frgpup.y . . . 4  |-  ( ph  ->  A  e.  I )
3 frgpup.r . . . . 5  |-  .~  =  ( ~FG  `  I )
4 frgpup.u . . . . 5  |-  U  =  (varFGrp `  I )
53, 4vrgpval 16600 . . . 4  |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
61, 2, 5syl2anc 661 . . 3  |-  ( ph  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
76fveq2d 5870 . 2  |-  ( ph  ->  ( E `  ( U `  A )
)  =  ( E `
 [ <" <. A ,  (/) >. "> ]  .~  ) )
8 0ex 4577 . . . . . . . 8  |-  (/)  e.  _V
98prid1 4135 . . . . . . 7  |-  (/)  e.  { (/)
,  1o }
10 df2o3 7144 . . . . . . 7  |-  2o  =  { (/) ,  1o }
119, 10eleqtrri 2554 . . . . . 6  |-  (/)  e.  2o
12 opelxpi 5031 . . . . . 6  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  ->  <. A ,  (/) >.  e.  ( I  X.  2o ) )
132, 11, 12sylancl 662 . . . . 5  |-  ( ph  -> 
<. A ,  (/) >.  e.  ( I  X.  2o ) )
1413s1cld 12581 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e. Word  (
I  X.  2o ) )
15 frgpup.w . . . . 5  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
16 2on 7139 . . . . . . 7  |-  2o  e.  On
17 xpexg 6587 . . . . . . 7  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
181, 16, 17sylancl 662 . . . . . 6  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
19 wrdexg 12524 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
20 fvi 5925 . . . . . 6  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
2118, 19, 203syl 20 . . . . 5  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2215, 21syl5eq 2520 . . . 4  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
2314, 22eleqtrrd 2558 . . 3  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e.  W
)
24 frgpup.b . . . 4  |-  B  =  ( Base `  H
)
25 frgpup.n . . . 4  |-  N  =  ( invg `  H )
26 frgpup.t . . . 4  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
27 frgpup.h . . . 4  |-  ( ph  ->  H  e.  Grp )
28 frgpup.a . . . 4  |-  ( ph  ->  F : I --> B )
29 frgpup.g . . . 4  |-  G  =  (freeGrp `  I )
30 frgpup.x . . . 4  |-  X  =  ( Base `  G
)
31 frgpup.e . . . 4  |-  E  =  ran  ( g  e.  W  |->  <. [ g ]  .~  ,  ( H 
gsumg  ( T  o.  g
) ) >. )
3224, 25, 26, 27, 1, 28, 15, 3, 29, 30, 31frgpupval 16607 . . 3  |-  ( (
ph  /\  <" <. A ,  (/) >. ">  e.  W )  ->  ( E `  [ <" <. A ,  (/) >. "> ]  .~  )  =  ( H  gsumg  ( T  o.  <" <. A ,  (/) >. "> )
) )
3323, 32mpdan 668 . 2  |-  ( ph  ->  ( E `  [ <" <. A ,  (/) >. "> ]  .~  )  =  ( H  gsumg  ( T  o.  <" <. A ,  (/)
>. "> ) ) )
3424, 25, 26, 27, 1, 28frgpuptf 16603 . . . . . 6  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
35 s1co 12765 . . . . . 6  |-  ( (
<. A ,  (/) >.  e.  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  <" <. A ,  (/) >. "> )  =  <" ( T `
 <. A ,  (/) >.
) "> )
3613, 34, 35syl2anc 661 . . . . 5  |-  ( ph  ->  ( T  o.  <"
<. A ,  (/) >. "> )  =  <" ( T `  <. A ,  (/)
>. ) "> )
37 df-ov 6288 . . . . . . 7  |-  ( A T (/) )  =  ( T `  <. A ,  (/)
>. )
38 iftrue 3945 . . . . . . . . . 10  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
39 fveq2 5866 . . . . . . . . . 10  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
4038, 39sylan9eqr 2530 . . . . . . . . 9  |-  ( ( y  =  A  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  A ) )
41 fvex 5876 . . . . . . . . 9  |-  ( F `
 A )  e. 
_V
4240, 26, 41ovmpt2a 6418 . . . . . . . 8  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  -> 
( A T (/) )  =  ( F `  A ) )
432, 11, 42sylancl 662 . . . . . . 7  |-  ( ph  ->  ( A T (/) )  =  ( F `  A ) )
4437, 43syl5eqr 2522 . . . . . 6  |-  ( ph  ->  ( T `  <. A ,  (/) >. )  =  ( F `  A ) )
4544s1eqd 12579 . . . . 5  |-  ( ph  ->  <" ( T `
 <. A ,  (/) >.
) ">  =  <" ( F `  A ) "> )
4636, 45eqtrd 2508 . . . 4  |-  ( ph  ->  ( T  o.  <"
<. A ,  (/) >. "> )  =  <" ( F `  A ) "> )
4746oveq2d 6301 . . 3  |-  ( ph  ->  ( H  gsumg  ( T  o.  <"
<. A ,  (/) >. "> ) )  =  ( H  gsumg 
<" ( F `  A ) "> ) )
4828, 2ffvelrnd 6023 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  B )
4924gsumws1 15842 . . . 4  |-  ( ( F `  A )  e.  B  ->  ( H  gsumg 
<" ( F `  A ) "> )  =  ( F `  A ) )
5048, 49syl 16 . . 3  |-  ( ph  ->  ( H  gsumg 
<" ( F `  A ) "> )  =  ( F `  A ) )
5147, 50eqtrd 2508 . 2  |-  ( ph  ->  ( H  gsumg  ( T  o.  <"
<. A ,  (/) >. "> ) )  =  ( F `  A ) )
527, 33, 513eqtrd 2512 1  |-  ( ph  ->  ( E `  ( U `  A )
)  =  ( F `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ifcif 3939   {cpr 4029   <.cop 4033    |-> cmpt 4505    _I cid 4790   Oncon0 4878    X. cxp 4997   ran crn 5000    o. ccom 5003   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1oc1o 7124   2oc2o 7125   [cec 7310  Word cword 12501   <"cs1 12504   Basecbs 14493    gsumg cgsu 14699   Grpcgrp 15730   invgcminusg 15731   ~FG cefg 16539  freeGrpcfrgp 16540  varFGrpcvrgp 16541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-ec 7314  df-qs 7318  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  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 10978  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-word 12509  df-concat 12511  df-s1 12512  df-substr 12513  df-splice 12514  df-s2 12779  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-0g 14700  df-gsum 14701  df-imas 14766  df-qus 14767  df-mnd 15735  df-submnd 15790  df-frmd 15852  df-grp 15871  df-minusg 15872  df-efg 16542  df-frgp 16543  df-vrgp 16544
This theorem is referenced by:  frgpup3  16611
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