Theorem pgpfaclem3 20213

Description: Lemma for pgpfac 20214. (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 14606	. . 3 ⊢ ∅ ∈ Word 𝐶
2		pgpfac.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
3		ablgrp 19913	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		eqid 2762	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	4	dprd0 20161	. . . . . 6 ⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
6	2, 3, 5	3syl 19	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)}))
8		pgpfac.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
9	4	subg0cl 19258	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑈)
10	8, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑈)
11	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (0g‘𝐺) ∈ 𝑈)
12		eqid 2762	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝑈) = (𝐺 ↾s 𝑈)
13	12	subgbas 19254	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
14	8, 13	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
15	14	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺 ↾s 𝑈)))
16	12	subggrp 19253	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑈) ∈ Grp)
17	8, 16	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Grp)
18		grpmnd 19065	. . . . . . . . . 10 ⊢ ((𝐺 ↾s 𝑈) ∈ Grp → (𝐺 ↾s 𝑈) ∈ Mnd)
19		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘(𝐺 ↾s 𝑈)) = (Base‘(𝐺 ↾s 𝑈))
20		eqid 2762	. . . . . . . . . . 11 ⊢ (gEx‘(𝐺 ↾s 𝑈)) = (gEx‘(𝐺 ↾s 𝑈))
21	19, 20	gex1 19719	. . . . . . . . . 10 ⊢ ((𝐺 ↾s 𝑈) ∈ Mnd → ((gEx‘(𝐺 ↾s 𝑈)) = 1 ↔ (Base‘(𝐺 ↾s 𝑈)) ≈ 1o))
22	17, 18, 21	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ((gEx‘(𝐺 ↾s 𝑈)) = 1 ↔ (Base‘(𝐺 ↾s 𝑈)) ≈ 1o))
23	22	biimpa 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (Base‘(𝐺 ↾s 𝑈)) ≈ 1o)
24	15, 23	eqbrtrd 5131	. . . . . . 7 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 ≈ 1o)
25		en1eqsn 9248	. . . . . . 7 ⊢ (((0g‘𝐺) ∈ 𝑈 ∧ 𝑈 ≈ 1o) → 𝑈 = {(0g‘𝐺)})
26	11, 24, 25	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → 𝑈 = {(0g‘𝐺)})
27	26	eqeq2d 2773	. . . . 5 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g‘𝐺)}))
28	27	anbi2d 642	. . . 4 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ((𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g‘𝐺)})))
29	7, 28	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈))
30		breq2 5111	. . . . 5 ⊢ (𝑠 = ∅ → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd ∅))
31		oveq2 7424	. . . . . 6 ⊢ (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
32	31	eqeq1d 2764	. . . . 5 ⊢ (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
33	30, 32	anbi12d 644	. . . 4 ⊢ (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
34	33	rspcev 3579	. . 3 ⊢ ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
35	1, 29, 34	sylancr 599	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
36	12	subgabl 19964	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺 ↾s 𝑈) ∈ Abel)
37	2, 8, 36	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐺 ↾s 𝑈) ∈ Abel)
38		pgpfac.f	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
39		pgpfac.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
40	39	subgss 19251	. . . . . . . . 9 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
41	8, 40	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
42	38, 41	ssfid 9242	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Fin)
43	14, 42	eqeltrrd 2863	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝑈)) ∈ Fin)
44	19, 20	gexcl2 19717	. . . . . 6 ⊢ (((𝐺 ↾s 𝑈) ∈ Grp ∧ (Base‘(𝐺 ↾s 𝑈)) ∈ Fin) → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
45	17, 43, 44	syl2anc 596	. . . . 5 ⊢ (𝜑 → (gEx‘(𝐺 ↾s 𝑈)) ∈ ℕ)
46		eqid 2762	. . . . . 6 ⊢ (od‘(𝐺 ↾s 𝑈)) = (od‘(𝐺 ↾s 𝑈))
47	19, 20, 46	gexex 19981	. . . . 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 596	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
49	48	adantr 486	. . 3 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺 ↾s 𝑈))((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
50		eqid 2762	. . . . 5 ⊢ (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈))) = (mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))
51		eqid 2762	. . . . 5 ⊢ ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺 ↾s 𝑈)))‘{𝑥})
52		eqid 2762	. . . . 5 ⊢ (0g‘(𝐺 ↾s 𝑈)) = (0g‘(𝐺 ↾s 𝑈))
53		eqid 2762	. . . . 5 ⊢ (LSSum‘(𝐺 ↾s 𝑈)) = (LSSum‘(𝐺 ↾s 𝑈))
54		pgpfac.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
55		subgpgp 19725	. . . . . . 7 ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑈))
56	54, 8, 55	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝑈))
57	56	ad2antrr 739	. . . . 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 739	. . . . 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 739	. . . . 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 785	. . . . 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 783	. . . . 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 20210	. . . 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 743	. . . . 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 743	. . . . 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 743	. . . . 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 743	. . . . 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 743	. . . . 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 788	. . . . 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 789	. . . . . 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 18	. . . . . 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 2865	. . . . 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 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))))) → ((od‘(𝐺 ↾s 𝑈))‘𝑥) = (gEx‘(𝐺 ↾s 𝑈)))
75		simprl 783	. . . . 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 793	. . . . 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 794	. . . . . 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 2800	. . . . 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 20212	. . . 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 3171	. . 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 3171	. 2 ⊢ ((𝜑 ∧ (gEx‘(𝐺 ↾s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
82	35, 81	pm2.61dane 3044	1 ⊢ (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4282  {csn 4587   class class class wbr 5107  dom cdm 5659  ran crn 5660  ‘cfv 6537  (class class class)co 7416  1_oc1o 8451   ≈ cen 8952  Fincfn 8955  1c1 11128  ℕcn 12260  Word cword 14580  Basecbs 17305   ↾_s cress 17326  0_gc0g 17528  mrClscmrc 17671  Mndcmnd 18838  Grpcgrp 19058  SubGrpcsubg 19244  odcod 19652  gExcgex 19653   pGrp cpgp 19654  LSSumclsm 19762  Abelcabl 19909  CycGrpccyg 20005   DProd cdprd 20123

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-disj 5075  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-rpss 7727  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-tpos 8227  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-omul 8463  df-er 8699  df-ec 8701  df-qs 8705  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-inf 9416  df-oi 9485  df-dju 9909  df-card 9947  df-acn 9950  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-xnn0 12605  df-z 12619  df-uz 12891  df-q 13001  df-rp 13045  df-fz 13564  df-fzo 13712  df-fl 13855  df-mod 13933  df-seq 14068  df-exp 14128  df-fac 14340  df-bc 14369  df-hash 14397  df-word 14581  df-concat 14638  df-s1 14665  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-clim 15577  df-sum 15776  df-dvds 16347  df-gcd 16589  df-prm 16766  df-pc 16933  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-0g 17530  df-gsum 17531  df-mre 17674  df-mrc 17675  df-acs 17677  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-submnd 18893  df-grp 19061  df-minusg 19062  df-sbg 19063  df-mulg 19192  df-subg 19247  df-eqg 19249  df-ghm 19342  df-gim 19387  df-ga 19418  df-cntz 19445  df-oppg 19474  df-od 19656  df-gex 19657  df-pgp 19658  df-lsm 19764  df-pj1 19765  df-cmn 19910  df-abl 19911  df-cyg 20006  df-dprd 20125

This theorem is used by:  pgpfac  20214