Theorem frgpup2 18189

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

Step	Hyp	Ref	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 )
5	3, 4	vrgpval 18180	. . . 4 |- ( ( I e. V /\ A e. I ) -> ( U ` A ) = [ <" <. A , (/) >. "> ] .~ )
6	1, 2, 5	syl2anc 693	. . 3 |- ( ph -> ( U ` A ) = [ <" <. A , (/) >. "> ] .~ )
7	6	fveq2d 6195	. 2 |- ( ph -> ( E ` ( U ` A ) ) = ( E ` [ <" <. A , (/) >. "> ] .~ ) )
8		0ex 4790	. . . . . . . 8 |- (/) e. _V
9	8	prid1 4297	. . . . . . 7 |- (/) e. { (/) , 1o }
10		df2o3 7573	. . . . . . 7 |- 2o = { (/) , 1o }
11	9, 10	eleqtrri 2700	. . . . . 6 |- (/) e. 2o
12		opelxpi 5148	. . . . . 6 |- ( ( A e. I /\ (/) e. 2o ) -> <. A , (/) >. e. ( I X. 2o ) )
13	2, 11, 12	sylancl 694	. . . . 5 |- ( ph -> <. A , (/) >. e. ( I X. 2o ) )
14	13	s1cld 13383	. . . 4 |- ( ph -> <" <. A , (/) >. "> e. Word ( I X. 2o ) )
15		frgpup.w	. . . . 5 |- W = ( _I ` Word ( I X. 2o ) )
16		2on 7568	. . . . . . 7 |- 2o e. On
17		xpexg 6960	. . . . . . 7 |- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
18	1, 16, 17	sylancl 694	. . . . . 6 |- ( ph -> ( I X. 2o ) e. _V )
19		wrdexg 13315	. . . . . 6 |- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
20		fvi 6255	. . . . . 6 |- (Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
21	18, 19, 20	3syl 18	. . . . 5 |- ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
22	15, 21	syl5eq 2668	. . . 4 |- ( ph -> W = Word ( I X. 2o ) )
23	14, 22	eleqtrrd 2704	. . 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 ) ) >. )
32	24, 25, 26, 27, 1, 28, 15, 3, 29, 30, 31	frgpupval 18187	. . 3 |- ( ( ph /\ <" <. A , (/) >. "> e. W ) -> ( E ` [ <" <. A , (/) >. "> ] .~ ) = ( H gsumg ( T o. <" <. A , (/) >. "> ) ) )
33	23, 32	mpdan 702	. 2 |- ( ph -> ( E ` [ <" <. A , (/) >. "> ] .~ ) = ( H gsumg ( T o. <" <. A , (/) >. "> ) ) )
34	24, 25, 26, 27, 1, 28	frgpuptf 18183	. . . . . 6 |- ( ph -> T : ( I X. 2o ) --> B )
35		s1co 13579	. . . . . 6 |- ( ( <. A , (/) >. e. ( I X. 2o ) /\ T : ( I X. 2o ) --> B ) -> ( T o. <" <. A , (/) >. "> ) = <" ( T ` <. A , (/) >. ) "> )
36	13, 34, 35	syl2anc 693	. . . . 5 |- ( ph -> ( T o. <" <. A , (/) >. "> ) = <" ( T ` <. A , (/) >. ) "> )
37		df-ov 6653	. . . . . . 7 |- ( A T (/) ) = ( T ` <. A , (/) >. )
38		iftrue 4092	. . . . . . . . . 10 |- ( z = (/) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` y ) )
39		fveq2 6191	. . . . . . . . . 10 |- ( y = A -> ( F ` y ) = ( F ` A ) )
40	38, 39	sylan9eqr 2678	. . . . . . . . 9 |- ( ( y = A /\ z = (/) ) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` A ) )
41		fvex 6201	. . . . . . . . 9 |- ( F ` A ) e. _V
42	40, 26, 41	ovmpt2a 6791	. . . . . . . 8 |- ( ( A e. I /\ (/) e. 2o ) -> ( A T (/) ) = ( F ` A ) )
43	2, 11, 42	sylancl 694	. . . . . . 7 |- ( ph -> ( A T (/) ) = ( F ` A ) )
44	37, 43	syl5eqr 2670	. . . . . 6 |- ( ph -> ( T ` <. A , (/) >. ) = ( F ` A ) )
45	44	s1eqd 13381	. . . . 5 |- ( ph -> <" ( T ` <. A , (/) >. ) "> = <" ( F ` A ) "> )
46	36, 45	eqtrd 2656	. . . 4 |- ( ph -> ( T o. <" <. A , (/) >. "> ) = <" ( F ` A ) "> )
47	46	oveq2d 6666	. . 3 |- ( ph -> ( H gsumg ( T o. <" <. A , (/) >. "> ) ) = ( H gsumg <" ( F ` A ) "> ) )
48	28, 2	ffvelrnd 6360	. . . 4 |- ( ph -> ( F ` A ) e. B )
49	24	gsumws1 17376	. . . 4 |- ( ( F ` A ) e. B -> ( H gsumg <" ( F ` A ) "> ) = ( F ` A ) )
50	48, 49	syl 17	. . 3 |- ( ph -> ( H gsumg <" ( F ` A ) "> ) = ( F ` A ) )
51	47, 50	eqtrd 2656	. 2 |- ( ph -> ( H gsumg ( T o. <" <. A , (/) >. "> ) ) = ( F ` A ) )
52	7, 33, 51	3eqtrd 2660	1 |- ( ph -> ( E ` ( U ` A ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   {cpr 4179   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   [cec 7740  Word cword 13291   <"cs1 13294   Basecbs 15857   gsum_g cgsu 16101   Grpcgrp 17422   invgcminusg 17423   ~_FG cefg 18119  freeGrpcfrgp 18120  var_FGrpcvrgp 18121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-gsum 16103  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-grp 17425  df-minusg 17426  df-efg 18122  df-frgp 18123  df-vrgp 18124

This theorem is referenced by:  frgpup3  18191