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Theorem frgpup2 17504
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 17495 . . . 4  |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
61, 2, 5syl2anc 673 . . 3  |-  ( ph  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
76fveq2d 5883 . 2  |-  ( ph  ->  ( E `  ( U `  A )
)  =  ( E `
 [ <" <. A ,  (/) >. "> ]  .~  ) )
8 0ex 4528 . . . . . . . 8  |-  (/)  e.  _V
98prid1 4071 . . . . . . 7  |-  (/)  e.  { (/)
,  1o }
10 df2o3 7213 . . . . . . 7  |-  2o  =  { (/) ,  1o }
119, 10eleqtrri 2548 . . . . . 6  |-  (/)  e.  2o
12 opelxpi 4871 . . . . . 6  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  ->  <. A ,  (/) >.  e.  ( I  X.  2o ) )
132, 11, 12sylancl 675 . . . . 5  |-  ( ph  -> 
<. A ,  (/) >.  e.  ( I  X.  2o ) )
1413s1cld 12795 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. ">  e. Word  (
I  X.  2o ) )
15 frgpup.w . . . . 5  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
16 2on 7208 . . . . . . 7  |-  2o  e.  On
17 xpexg 6612 . . . . . . 7  |-  ( ( I  e.  V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
181, 16, 17sylancl 675 . . . . . 6  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
19 wrdexg 12729 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
20 fvi 5937 . . . . . 6  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
2118, 19, 203syl 18 . . . . 5  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
2215, 21syl5eq 2517 . . . 4  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
2314, 22eleqtrrd 2552 . . 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 17502 . . 3  |-  ( (
ph  /\  <" <. A ,  (/) >. ">  e.  W )  ->  ( E `  [ <" <. A ,  (/) >. "> ]  .~  )  =  ( H  gsumg  ( T  o.  <" <. A ,  (/) >. "> )
) )
3323, 32mpdan 681 . 2  |-  ( ph  ->  ( E `  [ <" <. A ,  (/) >. "> ]  .~  )  =  ( H  gsumg  ( T  o.  <" <. A ,  (/)
>. "> ) ) )
3424, 25, 26, 27, 1, 28frgpuptf 17498 . . . . . 6  |-  ( ph  ->  T : ( I  X.  2o ) --> B )
35 s1co 12989 . . . . . 6  |-  ( (
<. A ,  (/) >.  e.  ( I  X.  2o )  /\  T : ( I  X.  2o ) --> B )  ->  ( T  o.  <" <. A ,  (/) >. "> )  =  <" ( T `
 <. A ,  (/) >.
) "> )
3613, 34, 35syl2anc 673 . . . . 5  |-  ( ph  ->  ( T  o.  <"
<. A ,  (/) >. "> )  =  <" ( T `  <. A ,  (/)
>. ) "> )
37 df-ov 6311 . . . . . . 7  |-  ( A T (/) )  =  ( T `  <. A ,  (/)
>. )
38 iftrue 3878 . . . . . . . . . 10  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
39 fveq2 5879 . . . . . . . . . 10  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
4038, 39sylan9eqr 2527 . . . . . . . . 9  |-  ( ( y  =  A  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  A ) )
41 fvex 5889 . . . . . . . . 9  |-  ( F `
 A )  e. 
_V
4240, 26, 41ovmpt2a 6446 . . . . . . . 8  |-  ( ( A  e.  I  /\  (/) 
e.  2o )  -> 
( A T (/) )  =  ( F `  A ) )
432, 11, 42sylancl 675 . . . . . . 7  |-  ( ph  ->  ( A T (/) )  =  ( F `  A ) )
4437, 43syl5eqr 2519 . . . . . 6  |-  ( ph  ->  ( T `  <. A ,  (/) >. )  =  ( F `  A ) )
4544s1eqd 12793 . . . . 5  |-  ( ph  ->  <" ( T `
 <. A ,  (/) >.
) ">  =  <" ( F `  A ) "> )
4636, 45eqtrd 2505 . . . 4  |-  ( ph  ->  ( T  o.  <"
<. A ,  (/) >. "> )  =  <" ( F `  A ) "> )
4746oveq2d 6324 . . 3  |-  ( ph  ->  ( H  gsumg  ( T  o.  <"
<. A ,  (/) >. "> ) )  =  ( H  gsumg 
<" ( F `  A ) "> ) )
4828, 2ffvelrnd 6038 . . . 4  |-  ( ph  ->  ( F `  A
)  e.  B )
4924gsumws1 16701 . . . 4  |-  ( ( F `  A )  e.  B  ->  ( H  gsumg 
<" ( F `  A ) "> )  =  ( F `  A ) )
5048, 49syl 17 . . 3  |-  ( ph  ->  ( H  gsumg 
<" ( F `  A ) "> )  =  ( F `  A ) )
5147, 50eqtrd 2505 . 2  |-  ( ph  ->  ( H  gsumg  ( T  o.  <"
<. A ,  (/) >. "> ) )  =  ( F `  A ) )
527, 33, 513eqtrd 2509 1  |-  ( ph  ->  ( E `  ( U `  A )
)  =  ( F `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   ifcif 3872   {cpr 3961   <.cop 3965    |-> cmpt 4454    _I cid 4749    X. cxp 4837   ran crn 4840    o. ccom 4843   Oncon0 5430   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1oc1o 7193   2oc2o 7194   [cec 7379  Word cword 12703   <"cs1 12706   Basecbs 15199    gsumg cgsu 15417   Grpcgrp 16747   invgcminusg 16748   ~FG cefg 17434  freeGrpcfrgp 17435  varFGrpcvrgp 17436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-s2 13003  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-0g 15418  df-gsum 15419  df-imas 15485  df-qus 15487  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-frmd 16711  df-grp 16751  df-minusg 16752  df-efg 17437  df-frgp 17438  df-vrgp 17439
This theorem is referenced by:  frgpup3  17506
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