Theorem ablfac2 17734

Description: Choose generators for each cyclic group in ablfac 17733. (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		wrdf 12683	. . . . . . . 8 |- ( s e. Word C -> s : ( 0..^ ( # ` s ) ) --> C )
2	1	ad2antlr 734	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : ( 0..^ ( # ` s ) ) --> C )
3		fdm 5738	. . . . . . 7 |- ( s : ( 0..^ ( # ` s ) ) --> C -> dom s = ( 0..^ ( # ` s ) ) )
4	2, 3	syl 17	. . . . . 6 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s = ( 0..^ ( # ` s ) ) )
5		fzofi 12194	. . . . . 6 |- ( 0..^ ( # ` s ) ) e. Fin
6	4, 5	syl6eqel 2539	. . . . 5 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s e. Fin )
7	4	feq2d 5720	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> ( s : dom s --> C <-> s : ( 0..^ ( # ` s ) ) --> C ) )
8	2, 7	mpbird 236	. . . . . . . . . 10 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : dom s --> C )
9	8	ffvelrnda 6027	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. C )
10		oveq2 6303	. . . . . . . . . . . 12 |- ( r = ( s ` k ) -> ( G ↾s r ) = ( G ↾s ( s ` k ) ) )
11	10	eleq1d 2515	. . . . . . . . . . 11 |- ( r = ( s ` k ) -> ( ( G ↾s r ) e. (CycGrp i^i ran pGrp ) <-> ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) ) )
12		ablfac.c	. . . . . . . . . . 11 |- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
13	11, 12	elrab2 3200	. . . . . . . . . 10 |- ( ( s ` k ) e. C <-> ( ( s ` k ) e. (SubGrp ` G ) /\ ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) ) )
14	13	simplbi 462	. . . . . . . . 9 |- ( ( s ` k ) e. C -> ( s ` k ) e. (SubGrp ` G ) )
15	9, 14	syl 17	. . . . . . . 8 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. (SubGrp ` G ) )
16		ablfac.b	. . . . . . . . 9 |- B = ( Base ` G )
17	16	subgss 16830	. . . . . . . 8 |- ( ( s ` k ) e. (SubGrp ` G ) -> ( s ` k ) C_ B )
18	15, 17	syl 17	. . . . . . 7 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) C_ B )
19		inss1 3654	. . . . . . . . . . 11 |- (CycGrp i^i ran pGrp ) C_ CycGrp
20	13	simprbi 466	. . . . . . . . . . . 12 |- ( ( s ` k ) e. C -> ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) )
21	9, 20	syl 17	. . . . . . . . . . 11 |- ( ( ( ( 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 ) )
22	19, 21	sseldi 3432	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( G ↾s ( s ` k ) ) e. CycGrp )
23		eqid 2453	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s ( s ` k ) ) ) = ( Base ` ( G ↾s ( s ` k ) ) )
24		eqid 2453	. . . . . . . . . . . 12 |- (.g ` ( G ↾s ( s ` k ) ) ) = (.g ` ( G ↾s ( s ` k ) ) )
25	23, 24	iscyg 17526	. . . . . . . . . . 11 |- ( ( 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 ) ) ) ) )
26	25	simprbi 466	. . . . . . . . . 10 |- ( ( 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 ) ) ) )
27	22, 26	syl 17	. . . . . . . . 9 |- ( ( ( ( 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 ) ) ) )
28		eqid 2453	. . . . . . . . . . . 12 |- ( G ↾s ( s ` k ) ) = ( G ↾s ( s ` k ) )
29	28	subgbas 16833	. . . . . . . . . . 11 |- ( ( s ` k ) e. (SubGrp ` G ) -> ( s ` k ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
30	15, 29	syl 17	. . . . . . . . . 10 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) = ( Base ` ( G ↾s ( s ` k ) ) ) )
31	30	rexeqdv 2996	. . . . . . . . 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 (.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 ) ) ) ) )
32	27, 31	mpbird 236	. . . . . . . 8 |- ( ( ( ( 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 ) ) ) )
33	15	ad2antrr 733	. . . . . . . . . . . . 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 ) -> ( s ` k ) e. (SubGrp ` G ) )
34		simpr 463	. . . . . . . . . . . . 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 e. ZZ )
35		simplr 763	. . . . . . . . . . . . 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 ) -> x e. ( s ` k ) )
36		ablfac2.m	. . . . . . . . . . . . . 14 |- .x. = (.g ` G )
37	36, 28, 24	subgmulg 16843	. . . . . . . . . . . . 13 |- ( ( ( s ` k ) e. (SubGrp ` G ) /\ n e. ZZ /\ x e. ( s ` k ) ) -> ( n .x. x ) = ( n (.g ` ( G ↾s ( s ` k ) ) ) x ) )
38	33, 34, 35, 37	syl3anc 1269	. . . . . . . . . . . 12 |- ( ( ( ( ( ( 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 ) )
39	38	mpteq2dva 4492	. . . . . . . . . . 11 |- ( ( ( ( ( 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 ) ) )
40	39	rneqd 5065	. . . . . . . . . 10 |- ( ( ( ( ( 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 ) ) )
41	30	adantr 467	. . . . . . . . . 10 |- ( ( ( ( ( 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 ) ) ) )
42	40, 41	eqeq12d 2468	. . . . . . . . 9 |- ( ( ( ( ( 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 ) ) ) ) )
43	42	rexbidva 2900	. . . . . . . 8 |- ( ( ( ( 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 ) ) ) ) )
44	32, 43	mpbird 236	. . . . . . 7 |- ( ( ( ( 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 ) )
45		ssrexv 3496	. . . . . . 7 |- ( ( 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 ) ) )
46	18, 44, 45	sylc 62	. . . . . 6 |- ( ( ( ( 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 ) )
47	46	ralrimiva 2804	. . . . 5 |- ( ( ( 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 ) )
48		oveq2 6303	. . . . . . . . 9 |- ( x = ( w ` k ) -> ( n .x. x ) = ( n .x. ( w ` k ) ) )
49	48	mpteq2dv 4493	. . . . . . . 8 |- ( x = ( w ` k ) -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
50	49	rneqd 5065	. . . . . . 7 |- ( x = ( w ` k ) -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
51	50	eqeq1d 2455	. . . . . 6 |- ( x = ( w ` k ) -> ( ran ( n e. ZZ |-> ( n .x. x ) ) = ( s ` k ) <-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) ) )
52	51	ac6sfi 7820	. . . . 5 |- ( ( 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 ) ) )
53	6, 47, 52	syl2anc 667	. . . 4 |- ( ( ( 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 ) ) )
54		simprl 765	. . . . . . . . 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 : dom s --> B )
55	4	adantr 467	. . . . . . . . . 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 ) ) ) -> dom s = ( 0..^ ( # ` s ) ) )
56	55	feq2d 5720	. . . . . . . . 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 : dom s --> B <-> w : ( 0..^ ( # ` s ) ) --> B ) )
57	54, 56	mpbid 214	. . . . . . . 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 : ( 0..^ ( # ` s ) ) --> B )
58		iswrdi 12682	. . . . . . . 8 |- ( w : ( 0..^ ( # ` s ) ) --> B -> w e. Word B )
59	57, 58	syl 17	. . . . . . 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 )
60		fdm 5738	. . . . . . . . . . . . . 14 |- ( w : ( 0..^ ( # ` s ) ) --> B -> dom w = ( 0..^ ( # ` s ) ) )
61	57, 60	syl 17	. . . . . . . . . . . . 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 ) ) ) -> dom w = ( 0..^ ( # ` s ) ) )
62	61, 55	eqtr4d 2490	. . . . . . . . . . . 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 w = dom s )
63	62	eleq2d 2516	. . . . . . . . . . 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 <-> j e. dom s ) )
64	63	biimpa 487	. . . . . . . . . 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 ) ) ) /\ j e. dom w ) -> j e. dom s )
65		simprr 767	. . . . . . . . . . . 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 ) ) ) -> A. k e. dom s ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) )
66		simpl 459	. . . . . . . . . . . . . . . . . 18 |- ( ( k = j /\ n e. ZZ ) -> k = j )
67	66	fveq2d 5874	. . . . . . . . . . . . . . . . 17 |- ( ( k = j /\ n e. ZZ ) -> ( w ` k ) = ( w ` j ) )
68	67	oveq2d 6311	. . . . . . . . . . . . . . . 16 |- ( ( k = j /\ n e. ZZ ) -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
69	68	mpteq2dva 4492	. . . . . . . . . . . . . . 15 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
70	69	rneqd 5065	. . . . . . . . . . . . . 14 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
71		fveq2 5870	. . . . . . . . . . . . . 14 |- ( k = j -> ( s ` k ) = ( s ` j ) )
72	70, 71	eqeq12d 2468	. . . . . . . . . . . . 13 |- ( k = j -> ( ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) <-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) = ( s ` j ) ) )
73	72	rspccva 3151	. . . . . . . . . . . 12 |- ( ( 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 ) )
74	65, 73	sylan 474	. . . . . . . . . . 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 s ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) = ( s ` j ) )
75	8	adantr 467	. . . . . . . . . . . 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 : dom s --> C )
76	75	ffvelrnda 6027	. . . . . . . . . . 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 s ) -> ( s ` j ) e. C )
77	74, 76	eqeltrd 2531	. . . . . . . . . 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 ) ) ) /\ j e. dom s ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) e. C )
78	64, 77	syldan 473	. . . . . . . . 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 ) ) ) /\ j e. dom w ) -> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) e. C )
79		ablfac2.s	. . . . . . . . . 10 |- S = ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
80		fveq2 5870	. . . . . . . . . . . . . 14 |- ( k = j -> ( w ` k ) = ( w ` j ) )
81	80	oveq2d 6311	. . . . . . . . . . . . 13 |- ( k = j -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
82	81	mpteq2dv 4493	. . . . . . . . . . . 12 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
83	82	rneqd 5065	. . . . . . . . . . 11 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
84	83	cbvmptv 4498	. . . . . . . . . 10 |- ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) ) = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
85	79, 84	eqtri 2475	. . . . . . . . 9 |- S = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
86	78, 85	fmptd 6051	. . . . . . . 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 ) ) ) -> S : dom w --> C )
87		simprl 765	. . . . . . . . . 10 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> G dom DProd s )
88	87	adantr 467	. . . . . . . . 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 ) ) ) -> G dom DProd s )
89	62	raleqdv 2995	. . . . . . . . . . . . 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 ) ) ) -> ( 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 ) ) )
90	65, 89	mpbird 236	. . . . . . . . . . . 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 ) ) ) -> A. k e. dom w ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( s ` k ) )
91		mpteq12 4485	. . . . . . . . . . . 12 |- ( ( 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 ) ) )
92	62, 90, 91	syl2anc 667	. . . . . . . . . . 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 ) ) ) -> ( k e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) ) = ( k e. dom s |-> ( s ` k ) ) )
93	79, 92	syl5eq 2499	. . . . . . . . . 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 = ( k e. dom s |-> ( s ` k ) ) )
94		dprdf 17650	. . . . . . . . . . . 12 |- ( G dom DProd s -> s : dom s --> (SubGrp ` G ) )
95	88, 94	syl 17	. . . . . . . . . . 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 : dom s --> (SubGrp ` G ) )
96	95	feqmptd 5923	. . . . . . . . . 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 = ( k e. dom s |-> ( s ` k ) ) )
97	93, 96	eqtr4d 2490	. . . . . . . . 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 = s )
98	88, 97	breqtrrd 4432	. . . . . . . 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 ) ) ) -> G dom DProd S )
99	97	oveq2d 6311	. . . . . . . . 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 ) ) ) -> ( G DProd S ) = ( G DProd s ) )
100		simplrr 772	. . . . . . . . 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 ) ) ) -> ( G DProd s ) = B )
101	99, 100	eqtrd 2487	. . . . . . . 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 ) ) ) -> ( G DProd S ) = B )
102	86, 98, 101	3jca 1189	. . . . . . 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 ) ) ) -> ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )
103	59, 102	jca 535	. . . . . 6 |- ( ( ( ( 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 ) ) )
104	103	ex 436	. . . . 5 |- ( ( ( 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 ) ) ) )
105	104	eximdv 1766	. . . 4 |- ( ( ( 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 ) ) ) )
106	53, 105	mpd 15	. . 3 |- ( ( ( 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 ) ) )
107		df-rex 2745	. . 3 |- ( 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 ) ) )
108	106, 107	sylibr 216	. 2 |- ( ( ( 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 ) )
109		ablfac.1	. . 3 |- ( ph -> G e. Abel )
110		ablfac.2	. . 3 |- ( ph -> B e. Fin )
111	16, 12, 109, 110	ablfac 17733	. 2 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )
112	108, 111	r19.29a 2934	1 |- ( ph -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 986   = wceq 1446   E.wex 1665   e. wcel 1889   A.wral 2739   E.wrex 2740   {crab 2743   i^i cin 3405   C_ wss 3406   class class class wbr 4405   |-> cmpt 4464   dom cdm 4837   ran crn 4838   -->wf 5581   ` cfv 5585  (class class class)co 6295   Fincfn 7574   0cc0 9544   ZZcz 10944  ..^cfzo 11922   #chash 12522  Word cword 12663   Basecbs 15133   ↾_s cress 15134   Grpcgrp 16681  ._gcmg 16684  SubGrpcsubg 16823   pGrp cpgp 17181   Abelcabl 17443  CycGrpccyg 17524   DProd cdprd 17637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-disj 4377  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-rpss 6576  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-er 7368  df-ec 7370  df-qs 7374  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-acn 8381  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-fz 11792  df-fzo 11923  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-word 12671  df-concat 12673  df-s1 12674  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-dvds 14318  df-gcd 14481  df-prm 14635  df-pc 14799  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-0g 15352  df-gsum 15353  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-submnd 16595  df-grp 16685  df-minusg 16686  df-sbg 16687  df-mulg 16688  df-subg 16826  df-eqg 16828  df-ghm 16893  df-gim 16935  df-ga 16956  df-cntz 16983  df-oppg 17009  df-od 17184  df-gex 17186  df-pgp 17188  df-lsm 17300  df-pj1 17301  df-cmn 17444  df-abl 17445  df-cyg 17525  df-dprd 17639

This theorem is referenced by:  dchrpt  24207