Theorem ablfac2 18488

Description: Choose generators for each cyclic group in ablfac 18487. (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 13310	. . . . . . . 8 |- ( s e. Word C -> s : ( 0..^ ( # ` s ) ) --> C )
2	1	ad2antlr 763	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : ( 0..^ ( # ` s ) ) --> C )
3		fdm 6051	. . . . . . 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 12773	. . . . . 6 |- ( 0..^ ( # ` s ) ) e. Fin
6	4, 5	syl6eqel 2709	. . . . 5 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s e. Fin )
7	4	feq2d 6031	. . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : dom s --> C )
9	8	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. C )
10		oveq2 6658	. . . . . . . . . . . 12 |- ( r = ( s ` k ) -> ( G ↾s r ) = ( G ↾s ( s ` k ) ) )
11	10	eleq1d 2686	. . . . . . . . . . 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 3366	. . . . . . . . . 10 |- ( ( s ` k ) e. C <-> ( ( s ` k ) e. (SubGrp ` G ) /\ ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) ) )
14	13	simplbi 476	. . . . . . . . 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 17595	. . . . . . . 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 3833	. . . . . . . . . . 11 |- (CycGrp i^i ran pGrp ) C_ CycGrp
20	13	simprbi 480	. . . . . . . . . . . 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 3601	. . . . . . . . . 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 2622	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s ( s ` k ) ) ) = ( Base ` ( G ↾s ( s ` k ) ) )
24		eqid 2622	. . . . . . . . . . . 12 |- (.g ` ( G ↾s ( s ` k ) ) ) = (.g ` ( G ↾s ( s ` k ) ) )
25	23, 24	iscyg 18281	. . . . . . . . . . 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 480	. . . . . . . . . 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 2622	. . . . . . . . . . . 12 |- ( G ↾s ( s ` k ) ) = ( G ↾s ( s ` k ) )
29	28	subgbas 17598	. . . . . . . . . . 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 3145	. . . . . . . . 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 247	. . . . . . . 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 762	. . . . . . . . . . . . 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 477	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . 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 17608	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 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 4744	. . . . . . . . . . 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 5353	. . . . . . . . . 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 481	. . . . . . . . . 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 2637	. . . . . . . . 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 3049	. . . . . . . 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 247	. . . . . . 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 3667	. . . . . . 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 65	. . . . . 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 2966	. . . . 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 6658	. . . . . . . . 9 |- ( x = ( w ` k ) -> ( n .x. x ) = ( n .x. ( w ` k ) ) )
49	48	mpteq2dv 4745	. . . . . . . 8 |- ( x = ( w ` k ) -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
50	49	rneqd 5353	. . . . . . 7 |- ( x = ( w ` k ) -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
51	50	eqeq1d 2624	. . . . . 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 8204	. . . . 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 693	. . . 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 794	. . . . . . . . 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 481	. . . . . . . . . 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 6031	. . . . . . . . 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 222	. . . . . . . 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 13309	. . . . . . . 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 6051	. . . . . . . . . . . . . 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 2659	. . . . . . . . . . . 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 2687	. . . . . . . . . . 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 501	. . . . . . . . . 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 796	. . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . 18 |- ( ( k = j /\ n e. ZZ ) -> k = j )
67	66	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( k = j /\ n e. ZZ ) -> ( w ` k ) = ( w ` j ) )
68	67	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( ( k = j /\ n e. ZZ ) -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
69	68	mpteq2dva 4744	. . . . . . . . . . . . . . 15 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
70	69	rneqd 5353	. . . . . . . . . . . . . 14 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
71		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = j -> ( s ` k ) = ( s ` j ) )
72	70, 71	eqeq12d 2637	. . . . . . . . . . . . 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 3308	. . . . . . . . . . . 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 488	. . . . . . . . . . 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 481	. . . . . . . . . . . 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 6359	. . . . . . . . . . 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 2701	. . . . . . . . . 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 487	. . . . . . . . 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 6191	. . . . . . . . . . . . . 14 |- ( k = j -> ( w ` k ) = ( w ` j ) )
81	80	oveq2d 6666	. . . . . . . . . . . . 13 |- ( k = j -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
82	81	mpteq2dv 4745	. . . . . . . . . . . 12 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
83	82	rneqd 5353	. . . . . . . . . . 11 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
84	83	cbvmptv 4750	. . . . . . . . . 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 2644	. . . . . . . . 9 |- S = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
86	78, 85	fmptd 6385	. . . . . . . 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 794	. . . . . . . . . 10 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> G dom DProd s )
88	87	adantr 481	. . . . . . . . 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 3144	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 4736	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 2668	. . . . . . . . . 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 18405	. . . . . . . . . . . 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 6249	. . . . . . . . . 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 2659	. . . . . . . . 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 4681	. . . . . . . 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 6666	. . . . . . . . 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 801	. . . . . . . . 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 2656	. . . . . . . 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 1242	. . . . . . 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 554	. . . . . 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 450	. . . . 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 1846	. . . 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 2918	. . 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 224	. 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 18487	. 2 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )
112	108, 111	r19.29a 3078	1 |- ( ph -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   ZZcz 11377  ..^cfzo 12465   #chash 13117  Word cword 13291   Basecbs 15857   ↾_s cress 15858   Grpcgrp 17422  ._gcmg 17540  SubGrpcsubg 17588   pGrp cpgp 17946   Abelcabl 18194  CycGrpccyg 18279   DProd cdprd 18392

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-inf2 8538  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-ghm 17658  df-gim 17701  df-ga 17723  df-cntz 17750  df-oppg 17776  df-od 17948  df-gex 17949  df-pgp 17950  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196  df-cyg 18280  df-dprd 18394

This theorem is referenced by:  dchrpt  24992