Theorem pgpfaclem3 19862

Description: Lemma for pgpfac 19863. (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 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
pgpfac.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac.f	⊢ (𝜑 → 𝐵 ∈ Fin)
pgpfac.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pgpfac.a	⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))

Assertion

Ref	Expression
pgpfaclem3	⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

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

Allowed substitution hints:   𝜑(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 14427	. . 3 ⊢ ∅ ∈ Word 𝐶
2		pgpfac.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
3		ablgrp 19567	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		eqid 2736	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	4	dprd0 19810	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
6	2, 3, 5	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
8		pgpfac.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
9	4	subg0cl 18936	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑈)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑈)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (0g‘𝐺) ∈ 𝑈)
12		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝑈) = (𝐺 ↾s 𝑈)
13	12	subgbas 18932	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
14	8, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
15	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
16	12	subggrp 18931	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑈) ∈ Grp)
17	8, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Grp)
18		grpmnd 18755	. . . . . . . . . 10 ⊢ ((𝐺 ↾s 𝑈) ∈ Grp → (𝐺 ↾s 𝑈) ∈ Mnd)
19		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘(𝐺 ↾s 𝑈)) = (Base‘(𝐺 ↾s 𝑈))
20		eqid 2736	. . . . . . . . . . 11 ⊢ (gEx‘(𝐺 ↾s 𝑈)) = (gEx‘(𝐺 ↾s 𝑈))
21	19, 20	gex1 19373	. . . . . . . . . 10 ⊢ ((𝐺 ↾s 𝑈) ∈ Mnd → ((gEx‘(𝐺 ↾s 𝑈)) = 1 ↔ (Base‘(𝐺 ↾s 𝑈)) ≈ 1o))
22	17, 18, 21	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((gEx‘(𝐺 ↾s 𝑈)) = 1 ↔ (Base‘(𝐺 ↾s 𝑈)) ≈ 1o))
23	22	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (Base‘(𝐺 ↾s 𝑈)) ≈ 1o)
24	15, 23	eqbrtrd 5127	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 ≈ 1o)
25		en1eqsn 9218	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑈 ∧ 𝑈 ≈ 1o) → 𝑈 = {(0g‘𝐺)})
26	11, 24, 25	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = {(0g‘𝐺)})
27	26	eqeq2d 2747	. . . . 5 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g‘𝐺)}))
28	27	anbi2d 629	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)})))
29	7, 28	mpbird 256	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈))
30		breq2 5109	. . . . 5 ⊢ (𝑠 = ∅ → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd ∅))
31		oveq2 7365	. . . . . 6 ⊢ (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
32	31	eqeq1d 2738	. . . . 5 ⊢ (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
33	30, 32	anbi12d 631	. . . 4 ⊢ (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
34	33	rspcev 3581	. . 3 ⊢ ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
35	1, 29, 34	sylancr 587	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
36	12	subgabl 19614	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ↾s 𝑈) ∈ Abel)
37	2, 8, 36	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Abel)
38		pgpfac.f	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
39		pgpfac.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
40	39	subgss 18929	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
41	8, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
42	38, 41	ssfid 9211	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Fin)
43	14, 42	eqeltrrd 2839	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝑈)) ∈ Fin)
44	19, 20	gexcl2 19371	. . . . . 6 ⊢ (((𝐺 ↾s 𝑈) ∈ Grp ∧ (Base‘(𝐺 ↾s 𝑈)) ∈ Fin) → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
45	17, 43, 44	syl2anc 584	. . . . 5 ⊢ (𝜑 → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
46		eqid 2736	. . . . . 6 ⊢ (od‘(𝐺 ↾s 𝑈)) = (od‘(𝐺 ↾s 𝑈))
47	19, 20, 46	gexex 19631	. . . . 5 ⊢ (((𝐺 ↾s 𝑈) ∈ Abel ∧ (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ) → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
48	37, 45, 47	syl2anc 584	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
49	48	adantr 481	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
50		eqid 2736	. . . . 5 ⊢ (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈))) = (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))
51		eqid 2736	. . . . 5 ⊢ ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})
52		eqid 2736	. . . . 5 ⊢ (0g‘(𝐺 ↾s 𝑈)) = (0g‘(𝐺 ↾s 𝑈))
53		eqid 2736	. . . . 5 ⊢ (LSSum‘(𝐺 ↾s 𝑈)) = (LSSum‘(𝐺 ↾s 𝑈))
54		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
55		subgpgp 19379	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑈))
56	54, 8, 55	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝑈))
57	56	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → 𝑃 pGrp (𝐺 ↾s 𝑈))
58	37	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → (𝐺 ↾s 𝑈) ∈ Abel)
59	43	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → (Base‘(𝐺 ↾s 𝑈)) ∈ Fin)
60		simprr 771	. . . . 5 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
61		simprl 769	. . . . 5 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → 𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)))
62	50, 51, 19, 46, 20, 52, 53, 57, 58, 59, 60, 61	pgpfac1 19859	. . . 4 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → ∃𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈))((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))
63		pgpfac.c	. . . . 5 ⊢ 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺 ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
64	2	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝐺 ∈ Abel)
65	54	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝑃 pGrp 𝐺)
66	38	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝐵 ∈ Fin)
67	8	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝑈 ∈ (SubGrp‘𝐺))
68		pgpfac.a	. . . . . 6 ⊢ (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
69	68	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡 ⊊ 𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
70		simpllr 774	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 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)
71		simplrl 775	. . . . . 6 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)))
72	67, 13	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 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)))
73	71, 72	eleqtrrd 2841	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝑥 ∈ 𝑈)
74		simplrr 776	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 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)))
75		simprl 769	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → 𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)))
76		simprrl 779	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 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}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))})
77		simprrr 780	. . . . . 6 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 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)))
78	77, 72	eqtr4d 2779	. . . . 5 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 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)
79	39, 63, 64, 65, 66, 67, 69, 12, 50, 46, 20, 52, 53, 70, 73, 74, 75, 76, 78	pgpfaclem2 19861	. . . 4 ⊢ ((((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺 ↾s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺 ↾s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})(LSSum‘(𝐺 ↾s 𝑈))𝑤) = (Base‘(𝐺 ↾s 𝑈))))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
80	62, 79	rexlimddv 3158	. . 3 ⊢ (((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺 ↾s 𝑈)) ∧ ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
81	49, 80	rexlimddv 3158	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
82	35, 81	pm2.61dane 3032	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407   ∩ cin 3909   ⊆ wss 3910   ⊊ wpss 3911  ∅c0 4282  {csn 4586   class class class wbr 5105  dom cdm 5633  ran crn 5634  ‘cfv 6496  (class class class)co 7357  1_oc1o 8405   ≈ cen 8880  Fincfn 8883  1c1 11052  ℕcn 12153  Word cword 14402  Basecbs 17083   ↾_s cress 17112  0_gc0g 17321  mrClscmrc 17463  Mndcmnd 18556  Grpcgrp 18748  SubGrpcsubg 18922  odcod 19306  gExcgex 19307   pGrp cpgp 19308  LSSumclsm 19416  Abelcabl 19563  CycGrpccyg 19654   DProd cdprd 19772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-rpss 7660  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-eqg 18927  df-ghm 19006  df-gim 19049  df-ga 19070  df-cntz 19097  df-oppg 19124  df-od 19310  df-gex 19311  df-pgp 19312  df-lsm 19418  df-pj1 19419  df-cmn 19564  df-abl 19565  df-cyg 19655  df-dprd 19774

This theorem is referenced by:  pgpfac  19863