Theorem ablfac2 17800

Description: Choose generators for each cyclic group in ablfac 17799. (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 12723	. . . . . . . 8 |- ( s e. Word C -> s : ( 0..^ ( # ` s ) ) --> C )
2	1	ad2antlr 741	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : ( 0..^ ( # ` s ) ) --> C )
3		fdm 5745	. . . . . . 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 12225	. . . . . 6 |- ( 0..^ ( # ` s ) ) e. Fin
6	4, 5	syl6eqel 2557	. . . . 5 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s e. Fin )
7	4	feq2d 5725	. . . . . . . . . . 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 240	. . . . . . . . . 10 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> s : dom s --> C )
9	8	ffvelrnda 6037	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. C )
10		oveq2 6316	. . . . . . . . . . . 12 |- ( r = ( s ` k ) -> ( G ↾s r ) = ( G ↾s ( s ` k ) ) )
11	10	eleq1d 2533	. . . . . . . . . . 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 3186	. . . . . . . . . 10 |- ( ( s ` k ) e. C <-> ( ( s ` k ) e. (SubGrp ` G ) /\ ( G ↾s ( s ` k ) ) e. (CycGrp i^i ran pGrp ) ) )
14	13	simplbi 467	. . . . . . . . 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 16896	. . . . . . . 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 3643	. . . . . . . . . . 11 |- (CycGrp i^i ran pGrp ) C_ CycGrp
20	13	simprbi 471	. . . . . . . . . . . 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 3416	. . . . . . . . . 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 2471	. . . . . . . . . . . 12 |- ( Base ` ( G ↾s ( s ` k ) ) ) = ( Base ` ( G ↾s ( s ` k ) ) )
24		eqid 2471	. . . . . . . . . . . 12 |- (.g ` ( G ↾s ( s ` k ) ) ) = (.g ` ( G ↾s ( s ` k ) ) )
25	23, 24	iscyg 17592	. . . . . . . . . . 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 471	. . . . . . . . . 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 2471	. . . . . . . . . . . 12 |- ( G ↾s ( s ` k ) ) = ( G ↾s ( s ` k ) )
29	28	subgbas 16899	. . . . . . . . . . 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 2980	. . . . . . . . 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 240	. . . . . . . 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 740	. . . . . . . . . . . . 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 468	. . . . . . . . . . . . 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 770	. . . . . . . . . . . . 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 16909	. . . . . . . . . . . . 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 1292	. . . . . . . . . . . 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 4482	. . . . . . . . . . 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 5068	. . . . . . . . . 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 472	. . . . . . . . . 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 2486	. . . . . . . . 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 2889	. . . . . . . 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 240	. . . . . . 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 3480	. . . . . . 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 61	. . . . . 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 2809	. . . . 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 6316	. . . . . . . . 9 |- ( x = ( w ` k ) -> ( n .x. x ) = ( n .x. ( w ` k ) ) )
49	48	mpteq2dv 4483	. . . . . . . 8 |- ( x = ( w ` k ) -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
50	49	rneqd 5068	. . . . . . 7 |- ( x = ( w ` k ) -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
51	50	eqeq1d 2473	. . . . . 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 7833	. . . . 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 673	. . . 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 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 ) ) ) -> w : dom s --> B )
55	4	adantr 472	. . . . . . . . . 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 5725	. . . . . . . . 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 215	. . . . . . . 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 12722	. . . . . . . 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 5745	. . . . . . . . . . . . . 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 2508	. . . . . . . . . . . 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 2534	. . . . . . . . . . 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 492	. . . . . . . . . 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 774	. . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . 18 |- ( ( k = j /\ n e. ZZ ) -> k = j )
67	66	fveq2d 5883	. . . . . . . . . . . . . . . . 17 |- ( ( k = j /\ n e. ZZ ) -> ( w ` k ) = ( w ` j ) )
68	67	oveq2d 6324	. . . . . . . . . . . . . . . 16 |- ( ( k = j /\ n e. ZZ ) -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
69	68	mpteq2dva 4482	. . . . . . . . . . . . . . 15 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
70	69	rneqd 5068	. . . . . . . . . . . . . 14 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
71		fveq2 5879	. . . . . . . . . . . . . 14 |- ( k = j -> ( s ` k ) = ( s ` j ) )
72	70, 71	eqeq12d 2486	. . . . . . . . . . . . 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 3135	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 472	. . . . . . . . . . . 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 6037	. . . . . . . . . . 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 2549	. . . . . . . . . 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 478	. . . . . . . . 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 5879	. . . . . . . . . . . . . 14 |- ( k = j -> ( w ` k ) = ( w ` j ) )
81	80	oveq2d 6324	. . . . . . . . . . . . 13 |- ( k = j -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
82	81	mpteq2dv 4483	. . . . . . . . . . . 12 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
83	82	rneqd 5068	. . . . . . . . . . 11 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
84	83	cbvmptv 4488	. . . . . . . . . 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 2493	. . . . . . . . 9 |- S = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
86	78, 85	fmptd 6061	. . . . . . . 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 772	. . . . . . . . . 10 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> G dom DProd s )
88	87	adantr 472	. . . . . . . . 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 2979	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 4475	. . . . . . . . . . . 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 673	. . . . . . . . . . 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 2517	. . . . . . . . . 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 17716	. . . . . . . . . . . 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 5932	. . . . . . . . . 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 2508	. . . . . . . . 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 4422	. . . . . . . 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 6324	. . . . . . . . 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 779	. . . . . . . . 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 2505	. . . . . . . 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 1210	. . . . . . 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 541	. . . . . 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 441	. . . . 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 1772	. . . 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 2762	. . 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 217	. 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 17799	. 2 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )
112	108, 111	r19.29a 2918	1 |- ( ph -> E. w e. Word B ( S : dom w --> C /\ G dom DProd S /\ ( G DProd S ) = B ) )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   i^i cin 3389   C_ wss 3390   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   0cc0 9557   ZZcz 10961  ..^cfzo 11942   #chash 12553  Word cword 12703   Basecbs 15199   ↾_s cress 15200   Grpcgrp 16747  ._gcmg 16750  SubGrpcsubg 16889   pGrp cpgp 17247   Abelcabl 17509  CycGrpccyg 17590   DProd cdprd 17703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-rpss 6590  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-eqg 16894  df-ghm 16959  df-gim 17001  df-ga 17022  df-cntz 17049  df-oppg 17075  df-od 17250  df-gex 17252  df-pgp 17254  df-lsm 17366  df-pj1 17367  df-cmn 17510  df-abl 17511  df-cyg 17591  df-dprd 17705

This theorem is referenced by:  dchrpt  24274