Theorem pgpfaclem3 19208

Description: Lemma for pgpfac 19209. (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 13892	. . 3 ⊢ ∅ ∈ Word 𝐶
2		pgpfac.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
3		ablgrp 18914	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		eqid 2824	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	4	dprd0 19156	. . . . . 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 483	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
8		pgpfac.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
9	4	subg0cl 18290	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑈)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑈)
11	10	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (0g‘𝐺) ∈ 𝑈)
12		eqid 2824	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝑈) = (𝐺 ↾s 𝑈)
13	12	subgbas 18286	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
14	8, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
15	14	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
16	12	subggrp 18285	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑈) ∈ Grp)
17	8, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Grp)
18		grpmnd 18113	. . . . . . . . . 10 ⊢ ((𝐺 ↾s 𝑈) ∈ Grp → (𝐺 ↾s 𝑈) ∈ Mnd)
19		eqid 2824	. . . . . . . . . . 11 ⊢ (Base‘(𝐺 ↾s 𝑈)) = (Base‘(𝐺 ↾s 𝑈))
20		eqid 2824	. . . . . . . . . . 11 ⊢ (gEx‘(𝐺 ↾s 𝑈)) = (gEx‘(𝐺 ↾s 𝑈))
21	19, 20	gex1 18719	. . . . . . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (Base‘(𝐺 ↾s 𝑈)) ≈ 1o)
24	15, 23	eqbrtrd 5091	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 ≈ 1o)
25		en1eqsn 8751	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑈 ∧ 𝑈 ≈ 1o) → 𝑈 = {(0g‘𝐺)})
26	11, 24, 25	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = {(0g‘𝐺)})
27	26	eqeq2d 2835	. . . . 5 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g‘𝐺)}))
28	27	anbi2d 630	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)})))
29	7, 28	mpbird 259	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈))
30		breq2 5073	. . . . 5 ⊢ (𝑠 = ∅ → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd ∅))
31		oveq2 7167	. . . . . 6 ⊢ (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
32	31	eqeq1d 2826	. . . . 5 ⊢ (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
33	30, 32	anbi12d 632	. . . 4 ⊢ (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
34	33	rspcev 3626	. . 3 ⊢ ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
35	1, 29, 34	sylancr 589	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
36	12	subgabl 18959	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ↾s 𝑈) ∈ Abel)
37	2, 8, 36	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Abel)
38		pgpfac.f	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
39		pgpfac.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
40	39	subgss 18283	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
41	8, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
42	38, 41	ssfid 8744	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Fin)
43	14, 42	eqeltrrd 2917	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝑈)) ∈ Fin)
44	19, 20	gexcl2 18717	. . . . . 6 ⊢ (((𝐺 ↾s 𝑈) ∈ Grp ∧ (Base‘(𝐺 ↾s 𝑈)) ∈ Fin) → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
45	17, 43, 44	syl2anc 586	. . . . 5 ⊢ (𝜑 → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
46		eqid 2824	. . . . . 6 ⊢ (od‘(𝐺 ↾s 𝑈)) = (od‘(𝐺 ↾s 𝑈))
47	19, 20, 46	gexex 18976	. . . . 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 586	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
49	48	adantr 483	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
50		eqid 2824	. . . . 5 ⊢ (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈))) = (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))
51		eqid 2824	. . . . 5 ⊢ ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})
52		eqid 2824	. . . . 5 ⊢ (0g‘(𝐺 ↾s 𝑈)) = (0g‘(𝐺 ↾s 𝑈))
53		eqid 2824	. . . . 5 ⊢ (LSSum‘(𝐺 ↾s 𝑈)) = (LSSum‘(𝐺 ↾s 𝑈))
54		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
55		subgpgp 18725	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑈))
56	54, 8, 55	syl2anc 586	. . . . . 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 19205	. . . 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 2919	. . . . 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 2862	. . . . 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 19207	. . . 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 3294	. . 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 3294	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
82	35, 81	pm2.61dane 3107	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  {crab 3145   ∩ cin 3938   ⊆ wss 3939   ⊊ wpss 3940  ∅c0 4294  {csn 4570   class class class wbr 5069  dom cdm 5558  ran crn 5559  ‘cfv 6358  (class class class)co 7159  1_oc1o 8098   ≈ cen 8509  Fincfn 8512  1c1 10541  ℕcn 11641  Word cword 13864  Basecbs 16486   ↾_s cress 16487  0_gc0g 16716  mrClscmrc 16857  Mndcmnd 17914  Grpcgrp 18106  SubGrpcsubg 18276  odcod 18655  gExcgex 18656   pGrp cpgp 18657  LSSumclsm 18762  Abelcabl 18910  CycGrpccyg 18999   DProd cdprd 19118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-disj 5035  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-rpss 7452  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-tpos 7895  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-omul 8110  df-er 8292  df-ec 8294  df-qs 8298  df-map 8411  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-sup 8909  df-inf 8910  df-oi 8977  df-dju 9333  df-card 9371  df-acn 9374  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-fac 13637  df-bc 13666  df-hash 13694  df-word 13865  df-concat 13926  df-s1 13953  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-sum 15046  df-dvds 15611  df-gcd 15847  df-prm 16019  df-pc 16177  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-0g 16718  df-gsum 16719  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-eqg 18281  df-ghm 18359  df-gim 18402  df-ga 18423  df-cntz 18450  df-oppg 18477  df-od 18659  df-gex 18660  df-pgp 18661  df-lsm 18764  df-pj1 18765  df-cmn 18911  df-abl 18912  df-cyg 19000  df-dprd 19120

This theorem is referenced by:  pgpfac  19209