Theorem pgpfaclem3 20135

Description: Lemma for pgpfac 20136. (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 14562	. . 3 ⊢ ∅ ∈ Word 𝐶
2		pgpfac.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
3		ablgrp 19835	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		eqid 2763	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	4	dprd0 20083	. . . . . 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 484	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
8		pgpfac.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
9	4	subg0cl 19186	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑈)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑈)
11	10	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (0g‘𝐺) ∈ 𝑈)
12		eqid 2763	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝑈) = (𝐺 ↾s 𝑈)
13	12	subgbas 19182	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
14	8, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
15	14	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
16	12	subggrp 19181	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑈) ∈ Grp)
17	8, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Grp)
18		grpmnd 18992	. . . . . . . . . 10 ⊢ ((𝐺 ↾s 𝑈) ∈ Grp → (𝐺 ↾s 𝑈) ∈ Mnd)
19		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘(𝐺 ↾s 𝑈)) = (Base‘(𝐺 ↾s 𝑈))
20		eqid 2763	. . . . . . . . . . 11 ⊢ (gEx‘(𝐺 ↾s 𝑈)) = (gEx‘(𝐺 ↾s 𝑈))
21	19, 20	gex1 19641	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (Base‘(𝐺 ↾s 𝑈)) ≈ 1o)
24	15, 23	eqbrtrd 5123	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 ≈ 1o)
25		en1eqsn 9219	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑈 ∧ 𝑈 ≈ 1o) → 𝑈 = {(0g‘𝐺)})
26	11, 24, 25	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = {(0g‘𝐺)})
27	26	eqeq2d 2774	. . . . 5 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g‘𝐺)}))
28	27	anbi2d 639	. . . 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 5105	. . . . 5 ⊢ (𝑠 = ∅ → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd ∅))
31		oveq2 7404	. . . . . 6 ⊢ (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
32	31	eqeq1d 2765	. . . . 5 ⊢ (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
33	30, 32	anbi12d 641	. . . 4 ⊢ (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
34	33	rspcev 3582	. . 3 ⊢ ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
35	1, 29, 34	sylancr 596	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
36	12	subgabl 19886	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ↾s 𝑈) ∈ Abel)
37	2, 8, 36	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Abel)
38		pgpfac.f	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
39		pgpfac.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
40	39	subgss 19179	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
41	8, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
42	38, 41	ssfid 9213	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Fin)
43	14, 42	eqeltrrd 2864	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝑈)) ∈ Fin)
44	19, 20	gexcl2 19639	. . . . . 6 ⊢ (((𝐺 ↾s 𝑈) ∈ Grp ∧ (Base‘(𝐺 ↾s 𝑈)) ∈ Fin) → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
45	17, 43, 44	syl2anc 593	. . . . 5 ⊢ (𝜑 → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
46		eqid 2763	. . . . . 6 ⊢ (od‘(𝐺 ↾s 𝑈)) = (od‘(𝐺 ↾s 𝑈))
47	19, 20, 46	gexex 19903	. . . . 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 593	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
49	48	adantr 484	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
50		eqid 2763	. . . . 5 ⊢ (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈))) = (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))
51		eqid 2763	. . . . 5 ⊢ ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})
52		eqid 2763	. . . . 5 ⊢ (0g‘(𝐺 ↾s 𝑈)) = (0g‘(𝐺 ↾s 𝑈))
53		eqid 2763	. . . . 5 ⊢ (LSSum‘(𝐺 ↾s 𝑈)) = (LSSum‘(𝐺 ↾s 𝑈))
54		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
55		subgpgp 19647	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑈))
56	54, 8, 55	syl2anc 593	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝑈))
57	56	ad2antrr 736	. . . . 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 736	. . . . 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 736	. . . . 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 782	. . . . 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 780	. . . . 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 20132	. . . 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 740	. . . . 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 740	. . . . 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 740	. . . . 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 740	. . . . 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 740	. . . . 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 785	. . . . 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 786	. . . . . 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 2866	. . . . 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 787	. . . . 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 780	. . . . 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 790	. . . . 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 791	. . . . . 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 2801	. . . . 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 20134	. . . 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 3170	. . 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 3170	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
82	35, 81	pm2.61dane 3045	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  ∃wrex 3087  {crab 3415   ∩ cin 3904   ⊆ wss 3905   ⊊ wpss 3906  ∅c0 4286  {csn 4583   class class class wbr 5101  dom cdm 5648  ran crn 5649  ‘cfv 6521  (class class class)co 7396  1_oc1o 8430   ≈ cen 8924  Fincfn 8927  1c1 11085  ℕcn 12220  Word cword 14536  Basecbs 17255   ↾_s cress 17276  0_gc0g 17478  mrClscmrc 17621  Mndcmnd 18778  Grpcgrp 18985  SubGrpcsubg 19172  odcod 19574  gExcgex 19575   pGrp cpgp 19576  LSSumclsm 19684  Abelcabl 19831  CycGrpccyg 19927   DProd cdprd 20045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-inf2 9594  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161  ax-pre-sup 11162

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-disj 5069  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-rpss 7706  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-er 8678  df-ec 8680  df-qs 8684  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-sup 9386  df-inf 9387  df-oi 9456  df-dju 9871  df-card 9909  df-acn 9912  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-div 11856  df-nn 12221  df-2 12290  df-3 12291  df-n0 12492  df-xnn0 12565  df-z 12579  df-uz 12850  df-q 12960  df-rp 13004  df-fz 13523  df-fzo 13670  df-fl 13812  df-mod 13890  df-seq 14025  df-exp 14085  df-fac 14297  df-bc 14326  df-hash 14354  df-word 14537  df-concat 14594  df-s1 14620  df-cj 15136  df-re 15137  df-im 15138  df-sqrt 15272  df-abs 15273  df-clim 15525  df-sum 15724  df-dvds 16297  df-gcd 16539  df-prm 16716  df-pc 16883  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-0g 17480  df-gsum 17481  df-mre 17624  df-mrc 17625  df-acs 17627  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-mhm 18827  df-submnd 18828  df-grp 18988  df-minusg 18989  df-sbg 18990  df-mulg 19120  df-subg 19175  df-eqg 19177  df-ghm 19264  df-gim 19309  df-ga 19340  df-cntz 19367  df-oppg 19396  df-od 19578  df-gex 19579  df-pgp 19580  df-lsm 19686  df-pj1 19687  df-cmn 19832  df-abl 19833  df-cyg 19928  df-dprd 20047

This theorem is referenced by:  pgpfac  20136