Theorem frgpup3lem 18190

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.

Step	Hyp	Ref	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 )
4	2, 3	ghmf 17664	. . 3 |- ( K e. ( G GrpHom H ) -> K : X --> B )
5		ffn 6045	. . 3 |- ( K : X --> B -> K Fn X )
6	1, 4, 5	3syl 18	. 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 ) ) >. )
16	3, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15	frgpup1 18188	. . 3 |- ( ph -> E e. ( G GrpHom H ) )
17	2, 3	ghmf 17664	. . 3 |- ( E e. ( G GrpHom H ) -> E : X --> B )
18		ffn 6045	. . 3 |- ( E : X --> B -> E Fn X )
19	16, 17, 18	3syl 18	. 2 |- ( ph -> E Fn X )
20		eqid 2622	. . . . . . . . 9 |- (freeMnd ` ( I X. 2o ) ) = (freeMnd ` ( I X. 2o ) )
21	14, 20, 13	frgpval 18171	. . . . . . . 8 |- ( I e. V -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
22	10, 21	syl 17	. . . . . . 7 |- ( ph -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
23		2on 7568	. . . . . . . . . . 11 |- 2o e. On
24		xpexg 6960	. . . . . . . . . . 11 |- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
25	10, 23, 24	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( I X. 2o ) e. _V )
26		wrdexg 13315	. . . . . . . . . 10 |- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
27		fvi 6255	. . . . . . . . . 10 |- (Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
28	25, 26, 27	3syl 18	. . . . . . . . 9 |- ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
29	12, 28	syl5eq 2668	. . . . . . . 8 |- ( ph -> W = Word ( I X. 2o ) )
30		eqid 2622	. . . . . . . . . 10 |- ( Base ` (freeMnd ` ( I X. 2o ) ) ) = ( Base ` (freeMnd ` ( I X. 2o ) ) )
31	20, 30	frmdbas 17389	. . . . . . . . 9 |- ( ( I X. 2o ) e. _V -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
32	25, 31	syl 17	. . . . . . . 8 |- ( ph -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
33	29, 32	eqtr4d 2659	. . . . . . 7 |- ( ph -> W = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
34		fvex 6201	. . . . . . . . 9 |- ( ~FG ` I ) e. _V
35	13, 34	eqeltri 2697	. . . . . . . 8 |- .~ e. _V
36	35	a1i 11	. . . . . . 7 |- ( ph -> .~ e. _V )
37		fvexd 6203	. . . . . . 7 |- ( ph -> (freeMnd ` ( I X. 2o ) ) e. _V )
38	22, 33, 36, 37	qusbas 16205	. . . . . 6 |- ( ph -> ( W /. .~ ) = ( Base ` G ) )
39	38, 2	syl6reqr 2675	. . . . 5 |- ( ph -> X = ( W /. .~ ) )
40		eqimss 3657	. . . . 5 |- ( X = ( W /. .~ ) -> X C_ ( W /. .~ ) )
41	39, 40	syl 17	. . . 4 |- ( ph -> X C_ ( W /. .~ ) )
42	41	sselda 3603	. . 3 |- ( ( ph /\ a e. X ) -> a e. ( W /. .~ ) )
43		eqid 2622	. . . 4 |- ( W /. .~ ) = ( W /. .~ )
44		fveq2 6191	. . . . 5 |- ( [ t ] .~ = a -> ( K ` [ t ] .~ ) = ( K ` a ) )
45		fveq2 6191	. . . . 5 |- ( [ t ] .~ = a -> ( E ` [ t ] .~ ) = ( E ` a ) )
46	44, 45	eqeq12d 2637	. . . 4 |- ( [ t ] .~ = a -> ( ( K ` [ t ] .~ ) = ( E ` [ t ] .~ ) <-> ( K ` a ) = ( E ` a ) ) )
47		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ t e. W ) -> t e. W )
48	29	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ t e. W ) -> W = Word ( I X. 2o ) )
49	47, 48	eleqtrd 2703	. . . . . . . . . . . . 13 |- ( ( ph /\ t e. W ) -> t e. Word ( I X. 2o ) )
50		wrdf 13310	. . . . . . . . . . . . 13 |- ( t e. Word ( I X. 2o ) -> t : ( 0..^ ( # ` t ) ) --> ( I X. 2o ) )
51	49, 50	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ t e. W ) -> t : ( 0..^ ( # ` t ) ) --> ( I X. 2o ) )
52	51	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> ( t ` n ) e. ( I X. 2o ) )
53		elxp2 5132	. . . . . . . . . . 11 |- ( ( t ` n ) e. ( I X. 2o ) <-> E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. )
54	52, 53	sylib 208	. . . . . . . . . 10 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. )
55		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( y = i -> ( F ` y ) = ( F ` i ) )
56	55	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( y = i -> ( N ` ( F ` y ) ) = ( N ` ( F ` i ) ) )
57	55, 56	ifeq12d 4106	. . . . . . . . . . . . . . . 16 |- ( y = i -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = if ( z = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) )
58		eqeq1 2626	. . . . . . . . . . . . . . . . 17 |- ( z = j -> ( z = (/) <-> j = (/) ) )
59	58	ifbid 4108	. . . . . . . . . . . . . . . 16 |- ( z = j -> if ( z = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) )
60		fvex 6201	. . . . . . . . . . . . . . . . 17 |- ( F ` i ) e. _V
61		fvex 6201	. . . . . . . . . . . . . . . . 17 |- ( N ` ( F ` i ) ) e. _V
62	60, 61	ifex 4156	. . . . . . . . . . . . . . . 16 |- if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) e. _V
63	57, 59, 8, 62	ovmpt2 6796	. . . . . . . . . . . . . . 15 |- ( ( i e. I /\ j e. 2o ) -> ( i T j ) = if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) )
64	63	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> ( i T j ) = if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) )
65		elpri 4197	. . . . . . . . . . . . . . . . 17 |- ( j e. { (/) , 1o } -> ( j = (/) \/ j = 1o ) )
66		df2o3 7573	. . . . . . . . . . . . . . . . 17 |- 2o = { (/) , 1o }
67	65, 66	eleq2s 2719	. . . . . . . . . . . . . . . 16 |- ( j e. 2o -> ( j = (/) \/ j = 1o ) )
68		frgpup3.e	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( K o. U ) = F )
69	68	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ i e. I ) -> ( K o. U ) = F )
70	69	fveq1d 6193	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. I ) -> ( ( K o. U ) ` i ) = ( F ` i ) )
71		frgpup.u	. . . . . . . . . . . . . . . . . . . . . . 23 |- U = (varFGrp ` I )
72	13, 71, 14, 2	vrgpf 18181	. . . . . . . . . . . . . . . . . . . . . 22 |- ( I e. V -> U : I --> X )
73	10, 72	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> U : I --> X )
74		fvco3 6275	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( U : I --> X /\ i e. I ) -> ( ( K o. U ) ` i ) = ( K ` ( U ` i ) ) )
75	73, 74	sylan 488	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. I ) -> ( ( K o. U ) ` i ) = ( K ` ( U ` i ) ) )
76	70, 75	eqtr3d 2658	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ i e. I ) -> ( F ` i ) = ( K ` ( U ` i ) ) )
77	76	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> ( F ` i ) = ( K ` ( U ` i ) ) )
78		iftrue 4092	. . . . . . . . . . . . . . . . . . 19 |- ( j = (/) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( F ` i ) )
79	78	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( F ` i ) )
80		simpr 477	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> j = (/) )
81	80	opeq2d 4409	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> <. i , j >. = <. i , (/) >. )
82	81	s1eqd 13381	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> <" <. i , j >. "> = <" <. i , (/) >. "> )
83	82	eceq1d 7783	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> [ <" <. i , j >. "> ] .~ = [ <" <. i , (/) >. "> ] .~ )
84	13, 71	vrgpval 18180	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( I e. V /\ i e. I ) -> ( U ` i ) = [ <" <. i , (/) >. "> ] .~ )
85	10, 84	sylan 488	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ i e. I ) -> ( U ` i ) = [ <" <. i , (/) >. "> ] .~ )
86	85	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> ( U ` i ) = [ <" <. i , (/) >. "> ] .~ )
87	83, 86	eqtr4d 2659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> [ <" <. i , j >. "> ] .~ = ( U ` i ) )
88	87	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> ( K ` [ <" <. i , j >. "> ] .~ ) = ( K ` ( U ` i ) ) )
89	77, 79, 88	3eqtr4d 2666	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
90	76	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. I ) -> ( N ` ( F ` i ) ) = ( N ` ( K ` ( U ` i ) ) ) )
91	1	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ i e. I ) -> K e. ( G GrpHom H ) )
92	73	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ i e. I ) -> ( U ` i ) e. X )
93		eqid 2622	. . . . . . . . . . . . . . . . . . . . . 22 |- ( invg ` G ) = ( invg ` G )
94	2, 93, 7	ghminv 17667	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( K e. ( G GrpHom H ) /\ ( U ` i ) e. X ) -> ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) = ( N ` ( K ` ( U ` i ) ) ) )
95	91, 92, 94	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ i e. I ) -> ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) = ( N ` ( K ` ( U ` i ) ) ) )
96	90, 95	eqtr4d 2659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ i e. I ) -> ( N ` ( F ` i ) ) = ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) )
97	96	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( N ` ( F ` i ) ) = ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) )
98		1n0 7575	. . . . . . . . . . . . . . . . . . . 20 |- 1o =/= (/)
99		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> j = 1o )
100	99	neeq1d 2853	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( j =/= (/) <-> 1o =/= (/) ) )
101	98, 100	mpbiri 248	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> j =/= (/) )
102		ifnefalse 4098	. . . . . . . . . . . . . . . . . . 19 |- ( j =/= (/) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( N ` ( F ` i ) ) )
103	101, 102	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( N ` ( F ` i ) ) )
104	99	opeq2d 4409	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> <. i , j >. = <. i , 1o >. )
105	104	s1eqd 13381	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> <" <. i , j >. "> = <" <. i , 1o >. "> )
106	105	eceq1d 7783	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> [ <" <. i , j >. "> ] .~ = [ <" <. i , 1o >. "> ] .~ )
107	13, 71, 14, 93	vrgpinv 18182	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( I e. V /\ i e. I ) -> ( ( invg ` G ) ` ( U ` i ) ) = [ <" <. i , 1o >. "> ] .~ )
108	10, 107	sylan 488	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ i e. I ) -> ( ( invg ` G ) ` ( U ` i ) ) = [ <" <. i , 1o >. "> ] .~ )
109	108	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( ( invg ` G ) ` ( U ` i ) ) = [ <" <. i , 1o >. "> ] .~ )
110	106, 109	eqtr4d 2659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> [ <" <. i , j >. "> ] .~ = ( ( invg ` G ) ` ( U ` i ) ) )
111	110	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( K ` [ <" <. i , j >. "> ] .~ ) = ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) )
112	97, 103, 111	3eqtr4d 2666	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
113	89, 112	jaodan 826	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ i e. I ) /\ ( j = (/) \/ j = 1o ) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
114	67, 113	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ i e. I ) /\ j e. 2o ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
115	114	anasss 679	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
116	64, 115	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> ( i T j ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
117		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( T ` <. i , j >. ) )
118		df-ov 6653	. . . . . . . . . . . . . . 15 |- ( i T j ) = ( T ` <. i , j >. )
119	117, 118	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( i T j ) )
120		s1eq 13380	. . . . . . . . . . . . . . . 16 |- ( ( t ` n ) = <. i , j >. -> <" ( t ` n ) "> = <" <. i , j >. "> )
121	120	eceq1d 7783	. . . . . . . . . . . . . . 15 |- ( ( t ` n ) = <. i , j >. -> [ <" ( t ` n ) "> ] .~ = [ <" <. i , j >. "> ] .~ )
122	121	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( t ` n ) = <. i , j >. -> ( K ` [ <" ( t ` n ) "> ] .~ ) = ( K ` [ <" <. i , j >. "> ] .~ ) )
123	119, 122	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( ( t ` n ) = <. i , j >. -> ( ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) <-> ( i T j ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) )
124	116, 123	syl5ibrcom 237	. . . . . . . . . . . 12 |- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> ( ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) )
125	124	rexlimdvva 3038	. . . . . . . . . . 11 |- ( ph -> ( E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) )
126	125	ad2antrr 762	. . . . . . . . . 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 ) "> ] .~ ) ) )
127	54, 126	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) )
128	127	mpteq2dva 4744	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> ( n e. ( 0..^ ( # ` t ) ) |-> ( T ` ( t ` n ) ) ) = ( n e. ( 0..^ ( # ` t ) ) |-> ( K ` [ <" ( t ` n ) "> ] .~ ) ) )
129	3, 7, 8, 9, 10, 11	frgpuptf 18183	. . . . . . . . . 10 |- ( ph -> T : ( I X. 2o ) --> B )
130	129	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> T : ( I X. 2o ) --> B )
131		fcompt 6400	. . . . . . . . 9 |- ( ( T : ( I X. 2o ) --> B /\ t : ( 0..^ ( # ` t ) ) --> ( I X. 2o ) ) -> ( T o. t ) = ( n e. ( 0..^ ( # ` t ) ) |-> ( T ` ( t ` n ) ) ) )
132	130, 51, 131	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> ( T o. t ) = ( n e. ( 0..^ ( # ` t ) ) |-> ( T ` ( t ` n ) ) ) )
133	52	s1cld 13383	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> <" ( t ` n ) "> e. Word ( I X. 2o ) )
134	29	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> W = Word ( I X. 2o ) )
135	133, 134	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> <" ( t ` n ) "> e. W )
136	14, 13, 12, 2	frgpeccl 18174	. . . . . . . . . 10 |- ( <" ( t ` n ) "> e. W -> [ <" ( t ` n ) "> ] .~ e. X )
137	135, 136	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ t e. W ) /\ n e. ( 0..^ ( # ` t ) ) ) -> [ <" ( t ` n ) "> ] .~ e. X )
138	51	feqmptd 6249	. . . . . . . . . . 11 |- ( ( ph /\ t e. W ) -> t = ( n e. ( 0..^ ( # ` t ) ) |-> ( t ` n ) ) )
139	10	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ t e. W ) -> I e. V )
140	139, 23, 24	sylancl 694	. . . . . . . . . . . 12 |- ( ( ph /\ t e. W ) -> ( I X. 2o ) e. _V )
141		eqid 2622	. . . . . . . . . . . . 13 |- (varFMnd ` ( I X. 2o ) ) = (varFMnd ` ( I X. 2o ) )
142	141	vrmdfval 17393	. . . . . . . . . . . 12 |- ( ( I X. 2o ) e. _V -> (varFMnd ` ( I X. 2o ) ) = ( w e. ( I X. 2o ) |-> <" w "> ) )
143	140, 142	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ t e. W ) -> (varFMnd ` ( I X. 2o ) ) = ( w e. ( I X. 2o ) |-> <" w "> ) )
144		s1eq 13380	. . . . . . . . . . 11 |- ( w = ( t ` n ) -> <" w "> = <" ( t ` n ) "> )
145	52, 138, 143, 144	fmptco 6396	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> ( (varFMnd ` ( I X. 2o ) ) o. t ) = ( n e. ( 0..^ ( # ` t ) ) |-> <" ( t ` n ) "> ) )
146		eqidd 2623	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) = ( w e. W |-> [ w ] .~ ) )
147		eceq1 7782	. . . . . . . . . 10 |- ( w = <" ( t ` n ) "> -> [ w ] .~ = [ <" ( t ` n ) "> ] .~ )
148	135, 145, 146, 147	fmptco 6396	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) o. ( (varFMnd ` ( I X. 2o ) ) o. t ) ) = ( n e. ( 0..^ ( # ` t ) ) |-> [ <" ( t ` n ) "> ] .~ ) )
149	1	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ t e. W ) -> K e. ( G GrpHom H ) )
150	149, 4	syl 17	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> K : X --> B )
151	150	feqmptd 6249	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> K = ( w e. X |-> ( K ` w ) ) )
152		fveq2 6191	. . . . . . . . 9 |- ( w = [ <" ( t ` n ) "> ] .~ -> ( K ` w ) = ( K ` [ <" ( t ` n ) "> ] .~ ) )
153	137, 148, 151, 152	fmptco 6396	. . . . . . . 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 ) "> ] .~ ) ) )
154	128, 132, 153	3eqtr4d 2666	. . . . . . 7 |- ( ( ph /\ t e. W ) -> ( T o. t ) = ( K o. ( ( w e. W |-> [ w ] .~ ) o. ( (varFMnd ` ( I X. 2o ) ) o. t ) ) ) )
155	154	oveq2d 6666	. . . . . 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 ) ) ) ) )
156	3, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15	frgpupval 18187	. . . . . 6 |- ( ( ph /\ t e. W ) -> ( E ` [ t ] .~ ) = ( H gsumg ( T o. t ) ) )
157		ghmmhm 17670	. . . . . . . 8 |- ( K e. ( G GrpHom H ) -> K e. ( G MndHom H ) )
158	149, 157	syl 17	. . . . . . 7 |- ( ( ph /\ t e. W ) -> K e. ( G MndHom H ) )
159	141	vrmdf 17395	. . . . . . . . . . 11 |- ( ( I X. 2o ) e. _V -> (varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) -->Word ( I X. 2o ) )
160	140, 159	syl 17	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> (varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) -->Word ( I X. 2o ) )
161	48	feq3d 6032	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> ( (varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> W <-> (varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) -->Word ( I X. 2o ) ) )
162	160, 161	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> (varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> W )
163		wrdco 13577	. . . . . . . . 9 |- ( ( t e. Word ( I X. 2o ) /\ (varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> W ) -> ( (varFMnd ` ( I X. 2o ) ) o. t ) e. Word W )
164	49, 162, 163	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> ( (varFMnd ` ( I X. 2o ) ) o. t ) e. Word W )
165	33	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ t e. W ) -> W = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
166	165	mpteq1d 4738	. . . . . . . . . . 11 |- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) = ( w e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) )
167		eqid 2622	. . . . . . . . . . . . 13 |- ( w e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) = ( w e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ )
168	20, 30, 14, 13, 167	frgpmhm 18178	. . . . . . . . . . . 12 |- ( I e. V -> ( w e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) e. ( (freeMnd ` ( I X. 2o ) ) MndHom G ) )
169	139, 168	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ t e. W ) -> ( w e. ( Base ` (freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) e. ( (freeMnd ` ( I X. 2o ) ) MndHom G ) )
170	166, 169	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) e. ( (freeMnd ` ( I X. 2o ) ) MndHom G ) )
171	30, 2	mhmf 17340	. . . . . . . . . 10 |- ( ( w e. W |-> [ w ] .~ ) e. ( (freeMnd ` ( I X. 2o ) ) MndHom G ) -> ( w e. W |-> [ w ] .~ ) : ( Base ` (freeMnd ` ( I X. 2o ) ) ) --> X )
172	170, 171	syl 17	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) : ( Base ` (freeMnd ` ( I X. 2o ) ) ) --> X )
173	165	feq2d 6031	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) : W --> X <-> ( w e. W |-> [ w ] .~ ) : ( Base ` (freeMnd ` ( I X. 2o ) ) ) --> X ) )
174	172, 173	mpbird 247	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) : W --> X )
175		wrdco 13577	. . . . . . . 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 )
176	164, 174, 175	syl2anc 693	. . . . . . 7 |- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) o. ( (varFMnd ` ( I X. 2o ) ) o. t ) ) e. Word X )
177	2	gsumwmhm 17382	. . . . . . 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 ) ) ) ) )
178	158, 176, 177	syl2anc 693	. . . . . 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 ) ) ) ) )
179	155, 156, 178	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ t e. W ) -> ( E ` [ t ] .~ ) = ( K ` ( G gsumg ( ( w e. W |-> [ w ] .~ ) o. ( (varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) )
180	20, 141	frmdgsum 17399	. . . . . . . . 9 |- ( ( ( I X. 2o ) e. _V /\ t e. Word ( I X. 2o ) ) -> ( (freeMnd ` ( I X. 2o ) ) gsumg ( (varFMnd ` ( I X. 2o ) ) o. t ) ) = t )
181	140, 49, 180	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> ( (freeMnd ` ( I X. 2o ) ) gsumg ( (varFMnd ` ( I X. 2o ) ) o. t ) ) = t )
182	181	fveq2d 6195	. . . . . . 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 ) )
183		wrdco 13577	. . . . . . . . . 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 ) )
184	49, 160, 183	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> ( (varFMnd ` ( I X. 2o ) ) o. t ) e. Word Word ( I X. 2o ) )
185	32	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ t e. W ) -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
186		wrdeq 13327	. . . . . . . . . 10 |- ( ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) -> Word ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word Word ( I X. 2o ) )
187	185, 186	syl 17	. . . . . . . . 9 |- ( ( ph /\ t e. W ) -> Word ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word Word ( I X. 2o ) )
188	184, 187	eleqtrrd 2704	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> ( (varFMnd ` ( I X. 2o ) ) o. t ) e. Word ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
189	30	gsumwmhm 17382	. . . . . . . 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 ) ) ) )
190	170, 188, 189	syl2anc 693	. . . . . . 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 ) ) ) )
191	12, 13	efger 18131	. . . . . . . . 9 |- .~ Er W
192	191	a1i 11	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> .~ Er W )
193		fvex 6201	. . . . . . . . . 10 |- ( _I ` Word ( I X. 2o ) ) e. _V
194	12, 193	eqeltri 2697	. . . . . . . . 9 |- W e. _V
195	194	a1i 11	. . . . . . . 8 |- ( ( ph /\ t e. W ) -> W e. _V )
196		eqid 2622	. . . . . . . 8 |- ( w e. W |-> [ w ] .~ ) = ( w e. W |-> [ w ] .~ )
197	192, 195, 196	divsfval 16207	. . . . . . 7 |- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) ` t ) = [ t ] .~ )
198	182, 190, 197	3eqtr3d 2664	. . . . . 6 |- ( ( ph /\ t e. W ) -> ( G gsumg ( ( w e. W |-> [ w ] .~ ) o. ( (varFMnd ` ( I X. 2o ) ) o. t ) ) ) = [ t ] .~ )
199	198	fveq2d 6195	. . . . 5 |- ( ( ph /\ t e. W ) -> ( K ` ( G gsumg ( ( w e. W |-> [ w ] .~ ) o. ( (varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) = ( K ` [ t ] .~ ) )
200	179, 199	eqtr2d 2657	. . . 4 |- ( ( ph /\ t e. W ) -> ( K ` [ t ] .~ ) = ( E ` [ t ] .~ ) )
201	43, 46, 200	ectocld 7814	. . 3 |- ( ( ph /\ a e. ( W /. .~ ) ) -> ( K ` a ) = ( E ` a ) )
202	42, 201	syldan 487	. 2 |- ( ( ph /\ a e. X ) -> ( K ` a ) = ( E ` a ) )
203	6, 19, 202	eqfnfvd 6314	1 |- ( ph -> K = E )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   {cpr 4179   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   o. ccom 5118   Oncon0 5723   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   [cec 7740   /.cqs 7741   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs1 13294   Basecbs 15857   gsum_g cgsu 16101   /._s cqus 16165   MndHom cmhm 17333  freeMndcfrmd 17384  var_FMndcvrmd 17385   Grpcgrp 17422   invgcminusg 17423   GrpHom cghm 17657   ~_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-xnn0 11364  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-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  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-mhm 17335  df-submnd 17336  df-frmd 17386  df-vrmd 17387  df-grp 17425  df-minusg 17426  df-ghm 17658  df-efg 18122  df-frgp 18123  df-vrgp 18124

This theorem is referenced by:  frgpup3  18191