Theorem pgpfaclem3 18482

Description: Lemma for pgpfac 18483. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)

Hypotheses

Ref	Expression
pgpfac.b	|- B = ( Base ` G )
pgpfac.c	|- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
pgpfac.g	|- ( ph -> G e. Abel )
pgpfac.p	|- ( ph -> P pGrp G )
pgpfac.f	|- ( ph -> B e. Fin )
pgpfac.u	|- ( ph -> U e. (SubGrp ` G ) )
pgpfac.a	|- ( ph -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )

Assertion

Ref	Expression
pgpfaclem3	|- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )

Distinct variable groups:   t, s, C   s, r, t, G   ph, t   B, s, t   U, r, s, t

Allowed substitution hints:   ph( s, r)   B( r)   C( r)   P( t, s, r)

Proof of Theorem pgpfaclem3

Dummy variables w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrd0 13330	. . 3 |- (/) e. Word C
2		pgpfac.g	. . . . . 6 |- ( ph -> G e. Abel )
3		ablgrp 18198	. . . . . 6 |- ( G e. Abel -> G e. Grp )
4		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
5	4	dprd0 18430	. . . . . 6 |- ( G e. Grp -> ( G dom DProd (/) /\ ( G DProd (/) ) = { ( 0g ` G ) } ) )
6	2, 3, 5	3syl 18	. . . . 5 |- ( ph -> ( G dom DProd (/) /\ ( G DProd (/) ) = { ( 0g ` G ) } ) )
7	6	adantr 481	. . . 4 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> ( G dom DProd (/) /\ ( G DProd (/) ) = { ( 0g ` G ) } ) )
8		pgpfac.u	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` G ) )
9	4	subg0cl 17602	. . . . . . . . 9 |- ( U e. (SubGrp ` G ) -> ( 0g ` G ) e. U )
10	8, 9	syl 17	. . . . . . . 8 |- ( ph -> ( 0g ` G ) e. U )
11	10	adantr 481	. . . . . . 7 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> ( 0g ` G ) e. U )
12		eqid 2622	. . . . . . . . . . 11 |- ( G ↾s U ) = ( G ↾s U )
13	12	subgbas 17598	. . . . . . . . . 10 |- ( U e. (SubGrp ` G ) -> U = ( Base ` ( G ↾s U ) ) )
14	8, 13	syl 17	. . . . . . . . 9 |- ( ph -> U = ( Base ` ( G ↾s U ) ) )
15	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> U = ( Base ` ( G ↾s U ) ) )
16	12	subggrp 17597	. . . . . . . . . . 11 |- ( U e. (SubGrp ` G ) -> ( G ↾s U ) e. Grp )
17	8, 16	syl 17	. . . . . . . . . 10 |- ( ph -> ( G ↾s U ) e. Grp )
18		grpmnd 17429	. . . . . . . . . 10 |- ( ( G ↾s U ) e. Grp -> ( G ↾s U ) e. Mnd )
19		eqid 2622	. . . . . . . . . . 11 |- ( Base ` ( G ↾s U ) ) = ( Base ` ( G ↾s U ) )
20		eqid 2622	. . . . . . . . . . 11 |- (gEx ` ( G ↾s U ) ) = (gEx ` ( G ↾s U ) )
21	19, 20	gex1 18006	. . . . . . . . . 10 |- ( ( G ↾s U ) e. Mnd -> ( (gEx ` ( G ↾s U ) ) = 1 <-> ( Base ` ( G ↾s U ) ) ~~ 1o ) )
22	17, 18, 21	3syl 18	. . . . . . . . 9 |- ( ph -> ( (gEx ` ( G ↾s U ) ) = 1 <-> ( Base ` ( G ↾s U ) ) ~~ 1o ) )
23	22	biimpa 501	. . . . . . . 8 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> ( Base ` ( G ↾s U ) ) ~~ 1o )
24	15, 23	eqbrtrd 4675	. . . . . . 7 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> U ~~ 1o )
25		en1eqsn 8190	. . . . . . 7 |- ( ( ( 0g ` G ) e. U /\ U ~~ 1o ) -> U = { ( 0g ` G ) } )
26	11, 24, 25	syl2anc 693	. . . . . 6 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> U = { ( 0g ` G ) } )
27	26	eqeq2d 2632	. . . . 5 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> ( ( G DProd (/) ) = U <-> ( G DProd (/) ) = { ( 0g ` G ) } ) )
28	27	anbi2d 740	. . . 4 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> ( ( G dom DProd (/) /\ ( G DProd (/) ) = U ) <-> ( G dom DProd (/) /\ ( G DProd (/) ) = { ( 0g ` G ) } ) ) )
29	7, 28	mpbird 247	. . 3 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> ( G dom DProd (/) /\ ( G DProd (/) ) = U ) )
30		breq2 4657	. . . . 5 |- ( s = (/) -> ( G dom DProd s <-> G dom DProd (/) ) )
31		oveq2 6658	. . . . . 6 |- ( s = (/) -> ( G DProd s ) = ( G DProd (/) ) )
32	31	eqeq1d 2624	. . . . 5 |- ( s = (/) -> ( ( G DProd s ) = U <-> ( G DProd (/) ) = U ) )
33	30, 32	anbi12d 747	. . . 4 |- ( s = (/) -> ( ( G dom DProd s /\ ( G DProd s ) = U ) <-> ( G dom DProd (/) /\ ( G DProd (/) ) = U ) ) )
34	33	rspcev 3309	. . 3 |- ( ( (/) e. Word C /\ ( G dom DProd (/) /\ ( G DProd (/) ) = U ) ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
35	1, 29, 34	sylancr 695	. 2 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) = 1 ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
36	12	subgabl 18241	. . . . . 6 |- ( ( G e. Abel /\ U e. (SubGrp ` G ) ) -> ( G ↾s U ) e. Abel )
37	2, 8, 36	syl2anc 693	. . . . 5 |- ( ph -> ( G ↾s U ) e. Abel )
38		pgpfac.f	. . . . . . . 8 |- ( ph -> B e. Fin )
39		pgpfac.b	. . . . . . . . . 10 |- B = ( Base ` G )
40	39	subgss 17595	. . . . . . . . 9 |- ( U e. (SubGrp ` G ) -> U C_ B )
41	8, 40	syl 17	. . . . . . . 8 |- ( ph -> U C_ B )
42		ssfi 8180	. . . . . . . 8 |- ( ( B e. Fin /\ U C_ B ) -> U e. Fin )
43	38, 41, 42	syl2anc 693	. . . . . . 7 |- ( ph -> U e. Fin )
44	14, 43	eqeltrrd 2702	. . . . . 6 |- ( ph -> ( Base ` ( G ↾s U ) ) e. Fin )
45	19, 20	gexcl2 18004	. . . . . 6 |- ( ( ( G ↾s U ) e. Grp /\ ( Base ` ( G ↾s U ) ) e. Fin ) -> (gEx ` ( G ↾s U ) ) e. NN )
46	17, 44, 45	syl2anc 693	. . . . 5 |- ( ph -> (gEx ` ( G ↾s U ) ) e. NN )
47		eqid 2622	. . . . . 6 |- ( od ` ( G ↾s U ) ) = ( od ` ( G ↾s U ) )
48	19, 20, 47	gexex 18256	. . . . 5 |- ( ( ( G ↾s U ) e. Abel /\ (gEx ` ( G ↾s U ) ) e. NN ) -> E. x e. ( Base ` ( G ↾s U ) ) ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) )
49	37, 46, 48	syl2anc 693	. . . 4 |- ( ph -> E. x e. ( Base ` ( G ↾s U ) ) ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) )
50	49	adantr 481	. . 3 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) -> E. x e. ( Base ` ( G ↾s U ) ) ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) )
51		eqid 2622	. . . . 5 |- (mrCls ` (SubGrp ` ( G ↾s U ) ) ) = (mrCls ` (SubGrp ` ( G ↾s U ) ) )
52		eqid 2622	. . . . 5 |- ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) = ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } )
53		eqid 2622	. . . . 5 |- ( 0g ` ( G ↾s U ) ) = ( 0g ` ( G ↾s U ) )
54		eqid 2622	. . . . 5 |- ( LSSum ` ( G ↾s U ) ) = ( LSSum ` ( G ↾s U ) )
55		pgpfac.p	. . . . . . 7 |- ( ph -> P pGrp G )
56		subgpgp 18012	. . . . . . 7 |- ( ( P pGrp G /\ U e. (SubGrp ` G ) ) -> P pGrp ( G ↾s U ) )
57	55, 8, 56	syl2anc 693	. . . . . 6 |- ( ph -> P pGrp ( G ↾s U ) )
58	57	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> P pGrp ( G ↾s U ) )
59	37	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> ( G ↾s U ) e. Abel )
60	44	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> ( Base ` ( G ↾s U ) ) e. Fin )
61		simprr 796	. . . . 5 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) )
62		simprl 794	. . . . 5 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> x e. ( Base ` ( G ↾s U ) ) )
63	51, 52, 19, 47, 20, 53, 54, 58, 59, 60, 61, 62	pgpfac1 18479	. . . 4 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> E. w e. (SubGrp ` ( G ↾s U ) ) ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) )
64		pgpfac.c	. . . . 5 |- C = { r e. (SubGrp ` G ) | ( G ↾s r ) e. (CycGrp i^i ran pGrp ) }
65	2	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> G e. Abel )
66	55	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> P pGrp G )
67	38	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> B e. Fin )
68	8	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> U e. (SubGrp ` G ) )
69		pgpfac.a	. . . . . 6 |- ( ph -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )
70	69	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> A. t e. (SubGrp ` G ) ( t C. U -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = t ) ) )
71		simpllr 799	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> (gEx ` ( G ↾s U ) ) =/= 1 )
72		simplrl 800	. . . . . 6 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> x e. ( Base ` ( G ↾s U ) ) )
73	68, 13	syl 17	. . . . . 6 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> U = ( Base ` ( G ↾s U ) ) )
74	72, 73	eleqtrrd 2704	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> x e. U )
75		simplrr 801	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) )
76		simprl 794	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> w e. (SubGrp ` ( G ↾s U ) ) )
77		simprrl 804	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } )
78		simprrr 805	. . . . . 6 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) )
79	78, 73	eqtr4d 2659	. . . . 5 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = U )
80	39, 64, 65, 66, 67, 68, 70, 12, 51, 47, 20, 53, 54, 71, 74, 75, 76, 77, 79	pgpfaclem2 18481	. . . 4 |- ( ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) /\ ( w e. (SubGrp ` ( G ↾s U ) ) /\ ( ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) i^i w ) = { ( 0g ` ( G ↾s U ) ) } /\ ( ( (mrCls ` (SubGrp ` ( G ↾s U ) ) ) ` { x } ) ( LSSum ` ( G ↾s U ) ) w ) = ( Base ` ( G ↾s U ) ) ) ) ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
81	63, 80	rexlimddv 3035	. . 3 |- ( ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) /\ ( x e. ( Base ` ( G ↾s U ) ) /\ ( ( od ` ( G ↾s U ) ) ` x ) = (gEx ` ( G ↾s U ) ) ) ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
82	50, 81	rexlimddv 3035	. 2 |- ( ( ph /\ (gEx ` ( G ↾s U ) ) =/= 1 ) -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )
83	35, 82	pm2.61dane 2881	1 |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   {csn 4177   class class class wbr 4653   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   Fincfn 7955   1c1 9937   NNcn 11020  Word cword 13291   Basecbs 15857   ↾_s cress 15858   0gc0g 16100  mrClscmrc 16243   Mndcmnd 17294   Grpcgrp 17422  SubGrpcsubg 17588   odcod 17944  gExcgex 17945   pGrp cpgp 17946   LSSumclsm 18049   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-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:  pgpfac  18483