Theorem ablfac2 16942

Description: Choose generators for each cyclic group in ablfac 16941. (Contributed by Mario Carneiro, 28-Apr-2016.)

Hypotheses

Ref	Expression
ablfac.b	|- B = ( Base ` G )
ablfac.c	|- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
ablfac.1	|- ( ph -> G e. Abel )
ablfac.2	|- ( ph -> B e. Fin )
ablfac2.m	|- .x. = (.g ` G )
ablfac2.s	|- S = ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )

Assertion

Ref	Expression
ablfac2	|- ( ph -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )

Distinct variable groups:   S, r   k, n, r, w, B   .x. , k, w   C, k, n, w   ph, k, n, w   k, G, n, r, w

Allowed substitution hints:   ph( r)   C( r)   S( w, k, n)   .x. ( n, r)

Proof of Theorem ablfac2

Dummy variables s x j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablfac.b	. . 3 |- B = ( Base ` G )
2		ablfac.c	. . 3 |- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
3		ablfac.1	. . 3 |- ( ph -> G e. Abel )
4		ablfac.2	. . 3 |- ( ph -> B e. Fin )
5	1, 2, 3, 4	ablfac 16941	. 2 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )
6		wrdf 12519	. . . . . . . . . 10 |- ( s e. Word C -> s : ( 0..^ ( # ` s ) ) --> C )
7	6	ad2antlr 726	. . . . . . . . 9 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : ( 0..^ ( # ` s ) ) --> C )
8		fdm 5735	. . . . . . . . 9 |- ( s : ( 0..^ ( # ` s ) ) --> C -> dom s = ( 0..^ ( # ` s ) ) )
9	7, 8	syl 16	. . . . . . . 8 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s = ( 0..^ ( # ` s ) ) )
10		fzofi 12052	. . . . . . . 8 |- ( 0..^ ( # ` s ) ) e. Fin
11	9, 10	syl6eqel 2563	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s e. Fin )
12	9	feq2d 5718	. . . . . . . . . . . . 13 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> ( s : dom s --> C <-> s : ( 0..^ ( # ` s ) ) --> C ) )
13	7, 12	mpbird 232	. . . . . . . . . . . 12 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : dom s --> C )
14	13	ffvelrnda 6021	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. C )
15		oveq2 6292	. . . . . . . . . . . . . 14 |- ( r = ( s ` k ) -> ( G ↾s r ) = ( G ↾s ( s ` k ) ) )
16	15	eleq1d 2536	. . . . . . . . . . . . 13 |- ( r = ( s ` k ) -> ( ( G ↾s r ) e. (CycGrp i^i ran pGrp ) <-> ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) ) )
17	16, 2	elrab2 3263	. . . . . . . . . . . 12 |- ( ( s ` k ) e. C <-> ( ( s ` k ) e. (SubGrp ` G ) /\ ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) ) )
18	17	simplbi 460	. . . . . . . . . . 11 |- ( ( s ` k ) e. C -> ( s ` k ) e. (SubGrp ` G ) )
19	14, 18	syl 16	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. (SubGrp ` G ) )
20	1	subgss 16007	. . . . . . . . . 10 |- ( ( s ` k ) e. (SubGrp ` G ) -> ( s ` k ) C_ B )
21	19, 20	syl 16	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) C_ B )
22		inss1 3718	. . . . . . . . . . . . 13 |- (CycGrp i^i ran pGrp ) C_ CycGrp
23	17	simprbi 464	. . . . . . . . . . . . . 14 |- ( ( s ` k ) e. C -> ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) )
24	14, 23	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) )
25	22, 24	sseldi 3502	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( G ↾s ( s ` k ) ) e. CycGrp )
26		eqid 2467	. . . . . . . . . . . . . 14 |- ( Base ` ( G ↾s ( s ` k ) ) ) = ( Base ` ( G ↾s ( s ` k ) ) )
27		eqid 2467	. . . . . . . . . . . . . 14 |- (.g ` ( G ↾s ( s ` k ) ) ) = (.g ` ( G ↾s ( s ` k ) ) )
28	26, 27	iscyg 16685	. . . . . . . . . . . . 13 |- ( ( G ↾s ( s ` k ) ) e. CycGrp <-> ( ( G ↾s ( s ` k ) ) e. Grp /\ E. x e. ( Base ` ( G ↾s ( s ` k ) ) ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) ) )
29	28	simprbi 464	. . . . . . . . . . . 12 |- ( ( G ↾s ( s ` k ) ) e. CycGrp -> E. x e. ( Base ` ( G ↾s ( s ` k ) ) ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
30	25, 29	syl 16	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> E. x e. ( Base ` ( G ↾s ( s ` k ) ) ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
31		eqid 2467	. . . . . . . . . . . . . 14 |- ( G ↾s ( s ` k ) ) = ( G ↾s ( s ` k ) )
32	31	subgbas 16010	. . . . . . . . . . . . 13 |- ( ( s ` k ) e. (SubGrp ` G ) -> ( s ` k ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
33	19, 32	syl 16	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
34	33	rexeqdv 3065	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( E. x e. ( s ` k ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) <-> E. x e. ( Base ` ( G ↾s ( s ` k ) ) ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) ) )
35	30, 34	mpbird 232	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> E. x e. ( s ` k ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
36	19	ad2antrr 725	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) /\ n e. ZZ ) -> ( s ` k ) e. (SubGrp ` G ) )
37		simpr 461	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) /\ n e. ZZ ) -> n e. ZZ )
38		simplr 754	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) /\ n e. ZZ ) -> x e. ( s ` k ) )
39		ablfac2.m	. . . . . . . . . . . . . . . 16 |- .x. = (.g ` G )
40	39, 31, 27	subgmulg 16020	. . . . . . . . . . . . . . 15 |- ( ( ( s ` k ) e. (SubGrp ` G ) /\ n e. ZZ /\ x e. ( s ` k ) ) -> ( n .x. x ) = ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) )
41	36, 37, 38, 40	syl3anc 1228	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) /\ n e. ZZ ) -> ( n .x. x ) = ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) )
42	41	mpteq2dva 4533	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) )
43	42	rneqd 5230	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) )
44	33	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) -> ( s ` k ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
45	43, 44	eqeq12d 2489	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) /\ x e. ( s ` k ) ) -> ( ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) <-> ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) ) )
46	45	rexbidva 2970	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( E. x e. ( s ` k ) ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) <-> E. x e. ( s ` k ) ran ( n e. ZZ |-> ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) ) = ( Base ` ( G ↾s ( s ` k ) ) ) ) )
47	35, 46	mpbird 232	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> E. x e. ( s ` k ) ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) )
48		ssrexv 3565	. . . . . . . . 9 |- ( ( s ` k ) C_ B -> ( E. x e. ( s ` k ) ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) -> E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) ) )
49	21, 47, 48	sylc 60	. . . . . . . 8 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) )
50	49	ralrimiva 2878	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> A. k e. dom s E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) )
51		oveq2 6292	. . . . . . . . . . 11 |- ( x = ( w ` k ) -> ( n .x. x ) = ( n .x. ( w ` k ) ) )
52	51	mpteq2dv 4534	. . . . . . . . . 10 |- ( x = ( w ` k ) -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
53	52	rneqd 5230	. . . . . . . . 9 |- ( x = ( w ` k ) -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
54	53	eqeq1d 2469	. . . . . . . 8 |- ( x = ( w ` k ) -> ( ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) <-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) )
55	54	ac6sfi 7764	. . . . . . 7 |- ( ( dom s e. Fin /\ A. k e. dom s E. x e. B ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) ) -> E. w ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) )
56	11, 50, 55	syl2anc 661	. . . . . 6 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> E. w ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) )
57		simprl 755	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> w : dom s --> B )
58	9	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> dom s = ( 0..^ ( # ` s ) ) )
59	58	feq2d 5718	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( w : dom s --> B <-> w : ( 0..^ ( # ` s ) ) --> B ) )
60	57, 59	mpbid 210	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> w : ( 0..^ ( # ` s ) ) --> B )
61		iswrdi 12518	. . . . . . . . . 10 |- ( w : ( 0..^ ( # ` s ) ) --> B -> w e. Word B )
62	60, 61	syl 16	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> w e. Word B )
63		fdm 5735	. . . . . . . . . . . . . . . 16 |- ( w : ( 0..^ ( # ` s ) ) --> B -> dom w = ( 0..^ ( # ` s ) ) )
64	60, 63	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> dom w = ( 0..^ ( # ` s ) ) )
65	64, 58	eqtr4d 2511	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> dom w = dom s )
66	65	eleq2d 2537	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( j e. dom w <-> j e. dom s ) )
67	66	biimpa 484	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) /\ j e. dom w ) -> j e. dom s )
68		simprr 756	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) )
69		simpl 457	. . . . . . . . . . . . . . . . . . . 20 |- ( ( k = j /\ n e. ZZ ) -> k = j )
70	69	fveq2d 5870	. . . . . . . . . . . . . . . . . . 19 |- ( ( k = j /\ n e. ZZ ) -> ( w ` k ) = ( w ` j ) )
71	70	oveq2d 6300	. . . . . . . . . . . . . . . . . 18 |- ( ( k = j /\ n e. ZZ ) -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
72	71	mpteq2dva 4533	. . . . . . . . . . . . . . . . 17 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
73	72	rneqd 5230	. . . . . . . . . . . . . . . 16 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
74		fveq2 5866	. . . . . . . . . . . . . . . 16 |- ( k = j -> ( s ` k ) = ( s ` j ) )
75	73, 74	eqeq12d 2489	. . . . . . . . . . . . . . 15 |- ( k = j -> ( ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) <-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) = ( s ` j ) ) )
76	75	rspccva 3213	. . . . . . . . . . . . . 14 |- ( ( A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) /\ j e. dom s ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) = ( s ` j ) )
77	68, 76	sylan 471	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) /\ j e. dom s ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) = ( s ` j ) )
78	13	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> s : dom s --> C )
79	78	ffvelrnda 6021	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) /\ j e. dom s ) -> ( s ` j ) e. C )
80	77, 79	eqeltrd 2555	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) /\ j e. dom s ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) e. C )
81	67, 80	syldan 470	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) /\ j e. dom w ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) e. C )
82		ablfac2.s	. . . . . . . . . . . 12 |- S = ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
83		fveq2 5866	. . . . . . . . . . . . . . . 16 |- ( k = j -> ( w ` k ) = ( w ` j ) )
84	83	oveq2d 6300	. . . . . . . . . . . . . . 15 |- ( k = j -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
85	84	mpteq2dv 4534	. . . . . . . . . . . . . 14 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
86	85	rneqd 5230	. . . . . . . . . . . . 13 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
87	86	cbvmptv 4538	. . . . . . . . . . . 12 |- ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) ) = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
88	82, 87	eqtri 2496	. . . . . . . . . . 11 |- S = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
89	81, 88	fmptd 6045	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> S : dom w --> C )
90		simprl 755	. . . . . . . . . . . 12 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> G dom DProd s )
91	90	adantr 465	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> G dom DProd s )
92	65	raleqdv 3064	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( A. k e. dom w ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) <-> A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) )
93	68, 92	mpbird 232	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> A. k e. dom w ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) )
94		mpteq12 4526	. . . . . . . . . . . . . 14 |- ( ( dom w = dom s /\ A. k e. dom w ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) -> ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) ) = ( k e. dom s |-> ( s ` k ) ) )
95	65, 93, 94	syl2anc 661	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) ) = ( k e. dom s |-> ( s ` k ) ) )
96	82, 95	syl5eq 2520	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> S = ( k e. dom s |-> ( s ` k ) ) )
97		dprdf 16842	. . . . . . . . . . . . . 14 |- ( G dom DProd s -> s : dom s --> (SubGrp ` G ) )
98	91, 97	syl 16	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> s : dom s --> (SubGrp ` G ) )
99	98	feqmptd 5920	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> s = ( k e. dom s |-> ( s ` k ) ) )
100	96, 99	eqtr4d 2511	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> S = s )
101	91, 100	breqtrrd 4473	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> G dom DProd S )
102	100	oveq2d 6300	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( G DProd S ) = ( G DProd s ) )
103		simplrr 760	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( G DProd s ) = B )
104	102, 103	eqtrd 2508	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( G DProd S ) = B )
105	89, 101, 104	3jca 1176	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )
106	62, 105	jca 532	. . . . . . . 8 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) ) -> ( w e. Word B /\ ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) )
107	106	ex 434	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> ( ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) -> ( w e. Word B /\ ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) ) )
108	107	eximdv 1686	. . . . . 6 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> ( E. w ( w : dom s --> B /\ A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) -> E. w ( w e. Word B /\ ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) ) )
109	56, 108	mpd 15	. . . . 5 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> E. w ( w e. Word B /\ ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) )
110		df-rex 2820	. . . . 5 |- ( E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) <-> E. w ( w e. Word B /\ ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) )
111	109, 110	sylibr 212	. . . 4 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )
112	111	ex 434	. . 3 |- ( ( ph /\ s e. Word C ) -> ( ( G dom DProd s /\ ( G DProd s ) = B ) -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) )
113	112	rexlimdva 2955	. 2 |- ( ph -> ( E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) ) )
114	5, 113	mpd 15	1 |- ( ph -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   i^i cin 3475   C_ wss 3476   class class class wbr 4447   |-> cmpt 4505   dom cdm 4999   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   Fincfn 7516   0cc0 9492   ZZcz 10864  ..^cfzo 11792   #chash 12373  Word cword 12500   Basecbs 14490   ↾_s cress 14491   Grpcgrp 15727  ._gcmg 15731  SubGrpcsubg 16000   pGrp cpgp 16357   Abelcabl 16605  CycGrpccyg 16683   DProd cdprd 16827

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 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 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-fal 1385  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-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  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-se 4839  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-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-rpss 6564  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-gsum 14698  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-eqg 16005  df-ghm 16070  df-gim 16112  df-ga 16133  df-cntz 16160  df-oppg 16186  df-od 16359  df-gex 16360  df-pgp 16361  df-lsm 16462  df-pj1 16463  df-cmn 16606  df-abl 16607  df-cyg 16684  df-dprd 16829

This theorem is referenced by:  dchrpt  23298