Theorem ablfac2 16612

Description: Choose generators for each cyclic group in ablfac 16611. (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 16611	. 2 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = B ) )
6		wrdf 12261	. . . . . . . . . 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 5584	. . . . . . . . 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 11817	. . . . . . . 8 |- ( 0..^ ( # ` s ) ) e. Fin
11	9, 10	syl6eqel 2531	. . . . . . 7 |- ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) -> dom s e. Fin )
12	9	feq2d 5568	. . . . . . . . . . . . 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 5864	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. Word C ) /\ ( G dom DProd s /\ ( G DProd s ) = B ) ) /\ k e. dom s ) -> ( s ` k ) e. C )
15		oveq2 6120	. . . . . . . . . . . . . 14 |- ( r = ( s ` k ) -> ( G ↾s r ) = ( G ↾s ( s ` k ) ) )
16	15	eleq1d 2509	. . . . . . . . . . . . 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 3140	. . . . . . . . . . . 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 15703	. . . . . . . . . 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 3591	. . . . . . . . . . . . 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 3375	. . . . . . . . . . . 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 2443	. . . . . . . . . . . . . 14 |- ( Base ` ( G ↾s ( s ` k ) ) ) = ( Base ` ( G ↾s ( s ` k ) ) )
27		eqid 2443	. . . . . . . . . . . . . 14 |- (.g ` ( G ↾s ( s ` k ) ) ) = (.g ` ( G ↾s ( s ` k ) ) )
28	26, 27	iscyg 16377	. . . . . . . . . . . . 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 2443	. . . . . . . . . . . . . 14 |- ( G ↾s ( s ` k ) ) = ( G ↾s ( s ` k ) )
32	31	subgbas 15706	. . . . . . . . . . . . 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 2945	. . . . . . . . . . 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 15716	. . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . 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 4399	. . . . . . . . . . . . 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 5088	. . . . . . . . . . . 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 2457	. . . . . . . . . . 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 2753	. . . . . . . . . 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 3438	. . . . . . . . 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 2820	. . . . . . 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 6120	. . . . . . . . . . 11 |- ( x = ( w ` k ) -> ( n .x. x ) = ( n .x. ( w ` k ) ) )
52	51	mpteq2dv 4400	. . . . . . . . . 10 |- ( x = ( w ` k ) -> ( n e. ZZ |-> ( n .x. x ) ) = ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
53	52	rneqd 5088	. . . . . . . . 9 |- ( x = ( w ` k ) -> ran ( n e. ZZ |-> ( n .x. x ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) )
54	53	eqeq1d 2451	. . . . . . . 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 7577	. . . . . . 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 5568	. . . . . . . . . . 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 12260	. . . . . . . . . 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 5584	. . . . . . . . . . . . . . . 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 2478	. . . . . . . . . . . . . 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 2510	. . . . . . . . . . . . 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 5716	. . . . . . . . . . . . . . . . . . 19 |- ( ( k = j /\ n e. ZZ ) -> ( w ` k ) = ( w ` j ) )
71	70	oveq2d 6128	. . . . . . . . . . . . . . . . . 18 |- ( ( k = j /\ n e. ZZ ) -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
72	71	mpteq2dva 4399	. . . . . . . . . . . . . . . . 17 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
73	72	rneqd 5088	. . . . . . . . . . . . . . . 16 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
74		fveq2 5712	. . . . . . . . . . . . . . . 16 |- ( k = j -> ( s ` k ) = ( s ` j ) )
75	73, 74	eqeq12d 2457	. . . . . . . . . . . . . . 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 3093	. . . . . . . . . . . . . 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 5864	. . . . . . . . . . . . 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 2517	. . . . . . . . . . . 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 5712	. . . . . . . . . . . . . . . 16 |- ( k = j -> ( w ` k ) = ( w ` j ) )
84	83	oveq2d 6128	. . . . . . . . . . . . . . 15 |- ( k = j -> ( n .x. ( w ` k ) ) = ( n .x. ( w ` j ) ) )
85	84	mpteq2dv 4400	. . . . . . . . . . . . . 14 |- ( k = j -> ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
86	85	rneqd 5088	. . . . . . . . . . . . 13 |- ( k = j -> ran ( n e. ZZ |-> ( n .x. ( w ` k ) ) ) = ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
87	86	cbvmptv 4404	. . . . . . . . . . . 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 2463	. . . . . . . . . . 11 |- S = ( j e. dom w |-> ran ( n e. ZZ |-> ( n .x. ( w ` j ) ) ) )
89	81, 88	fmptd 5888	. . . . . . . . . 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 2944	. . . . . . . . . . . . . . 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 4392	. . . . . . . . . . . . . 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 2487	. . . . . . . . . . . 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 16512	. . . . . . . . . . . . . 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 5765	. . . . . . . . . . . 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 2478	. . . . . . . . . . 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 4339	. . . . . . . . . 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 6128	. . . . . . . . . . 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 2475	. . . . . . . . . 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 1168	. . . . . . . . 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 1676	. . . . . 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 2742	. . . . 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 2862	. 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 965   = wceq 1369   E.wex 1586   e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740   i^i cin 3348   C_ wss 3349   class class class wbr 4313   e. cmpt 4371   dom cdm 4861   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112   Fincfn 7331   0cc0 9303   ZZcz 10667  ..^cfzo 11569   #chash 12124  Word cword 12242   Basecbs 14195   ↾_s cress 14196   Grpcgrp 15431  ._gcmg 15435  SubGrpcsubg 15696   pGrp cpgp 16051   Abelcabel 16299  CycGrpccyg 16375   DProd cdprd 16497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-disj 4284  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-rpss 6381  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-omul 6946  df-er 7122  df-ec 7124  df-qs 7128  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-sup 7712  df-oi 7745  df-card 8130  df-acn 8133  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-gsum 14402  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-mhm 15485  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-eqg 15701  df-ghm 15766  df-gim 15808  df-ga 15829  df-cntz 15856  df-oppg 15882  df-od 16053  df-gex 16054  df-pgp 16055  df-lsm 16156  df-pj1 16157  df-cmn 16300  df-abl 16301  df-cyg 16376  df-dprd 16499

This theorem is referenced by:  dchrpt  22628