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Theorem pgpfac1lem5 19194

Description: Lemma for pgpfac1 19195. (Contributed by Mario Carneiro, 27-Apr-2016.)

**Hypotheses**
Ref	Expression
pgpfac1.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
pgpfac1.s	⊢ 𝑆 = (𝐾‘{𝐴})
pgpfac1.b	⊢ 𝐵 = (Base‘𝐺)
pgpfac1.o	⊢ 𝑂 = (od‘𝐺)
pgpfac1.e	⊢ 𝐸 = (gEx‘𝐺)
pgpfac1.z	⊢ 0 = (0_g‘𝐺)
pgpfac1.l	⊢ ⊕ = (LSSum‘𝐺)
pgpfac1.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac1.n	⊢ (𝜑 → 𝐵 ∈ Fin)
pgpfac1.oe	⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
pgpfac1.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pgpfac1.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
pgpfac1.3	⊢ (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))

**Assertion**
Ref	Expression
pgpfac1lem5	⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

Distinct variable groups: 𝑡,𝑠, 0 𝐴,𝑠,𝑡 ⊕ ,𝑠,𝑡 𝑃,𝑠,𝑡 𝐵,𝑠,𝑡 𝐺,𝑠,𝑡 𝑈,𝑠,𝑡 𝑆,𝑠,𝑡 𝜑,𝑠,𝑡 𝐾,𝑠,𝑡 Allowed substitution hints: 𝐸(𝑡,𝑠) 𝑂(𝑡,𝑠)

Proof of Theorem pgpfac1lem5 Dummy variables 𝑏 𝑢 𝑣 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac1.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Fin)
2		pwfi 8803	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
3	1, 2	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 𝐵 ∈ Fin)
4	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝒫 𝐵 ∈ Fin)
5		pgpfac1.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgss 18272	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (SubGrp‘𝐺) → 𝑣 ⊆ 𝐵)
7	6	3ad2ant2 1131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ⊆ 𝐵)
8		velpw 4502	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝒫 𝐵 ↔ 𝑣 ⊆ 𝐵)
9	7, 8	sylibr 237	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ∈ 𝒫 𝐵)
10	9	rabssdv 4002	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ⊆ 𝒫 𝐵)
11	4, 10	ssfid 8725	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12		finnum 9361	. . . . . . 7 ⊢ ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ Fin → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ dom card)
13	11, 12	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ dom card)
14		pgpfac1.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝐾‘{𝐴})
15		pgpfac1.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ∈ Abel)
16		ablgrp 18903	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Grp)
18	5	subgacs 18305	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
19		acsmre 16915	. . . . . . . . . . . 12 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
20	17, 18, 19	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
21		pgpfac1.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
22	5	subgss 18272	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
24		pgpfac1.au	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
25	23, 24	sseldd 3916	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
26		pgpfac1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
27	26	mrcsncl 16875	. . . . . . . . . . 11 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
28	20, 25, 27	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
29	14, 28	eqeltrid 2894	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
30	29	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
31		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ⊊ 𝑈)
32	24	snssd 4702	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ⊆ 𝑈)
33	32, 23	sstrd 3925	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
34	20, 26, 33	mrcssidd 16888	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
35	34, 14	sseqtrrdi 3966	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
36		snssg 4678	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
37	25, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
38	35, 37	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
39	38	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝐴 ∈ 𝑆)
40		psseq1 4015	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝑣 ⊊ 𝑈 ↔ 𝑆 ⊊ 𝑈))
41		eleq2 2878	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑆))
42	40, 41	anbi12d 633	. . . . . . . . 9 ⊢ (𝑣 = 𝑆 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑆 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑆)))
43	42	rspcev 3571	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑆 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑆)) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
44	30, 31, 39, 43	syl12anc 835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
45		rabn0 4293	. . . . . . 7 ⊢ ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
46	44, 45	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅)
47		simpr1 1191	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)})
48		simpr2 1192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ≠ ∅)
49	11	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
50	49, 47	ssfid 8725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ∈ Fin)
51		simpr3 1193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → [⊊] Or 𝑢)
52		fin1a2lem10 9820	. . . . . . . . . 10 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ 𝑢)
53	48, 50, 51, 52	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → ∪ 𝑢 ∈ 𝑢)
54	47, 53	sseldd 3916	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → ∪ 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)})
55	54	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}))
56	55	alrimiv 1928	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∀𝑢((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}))
57		zornn0g 9916	. . . . . 6 ⊢ (({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ dom card ∧ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅ ∧ ∀𝑢((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)})) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤)
58	13, 46, 56, 57	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤)
59		psseq1 4015	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
60		eleq2 2878	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑤))
61	59, 60	anbi12d 633	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤)))
62	61	ralrab 3633	. . . . . 6 ⊢ (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
63	62	rexbii 3210	. . . . 5 ⊢ (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤 ↔ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
64	58, 63	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
65	64	ex 416	. . 3 ⊢ (𝜑 → (𝑆 ⊊ 𝑈 → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
66		pgpfac1.3	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
67		psseq1 4015	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝑣 ⊊ 𝑈 ↔ 𝑠 ⊊ 𝑈))
68		eleq2 2878	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑠))
69	67, 68	anbi12d 633	. . . . . 6 ⊢ (𝑣 = 𝑠 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
70	69	ralrab 3633	. . . . 5 ⊢ (∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠)))
71	66, 70	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))
72		r19.29 3216	. . . . 5 ⊢ ((∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ∧ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
73	69	elrab 3628	. . . . . . 7 ⊢ (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
74		ineq2 4133	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ∩ 𝑡) = (𝑆 ∩ 𝑣))
75	74	eqeq1d 2800	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ 𝑣))
77	76	eqeq1d 2800	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑣) = 𝑠))
78	75, 77	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑡 = 𝑣 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠)))
79	78	cbvrexvw 3397	. . . . . . . . 9 ⊢ (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ∃𝑣 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠))
80		simprrl 780	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) → 𝑠 ⊊ 𝑈)
81	80	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → 𝑠 ⊊ 𝑈)
82		simpr2 1192	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → (𝑆 ⊕ 𝑣) = 𝑠)
83	82	psseq1d 4020	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → ((𝑆 ⊕ 𝑣) ⊊ 𝑈 ↔ 𝑠 ⊊ 𝑈))
84	81, 83	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → (𝑆 ⊕ 𝑣) ⊊ 𝑈)
85		pssdif 4280	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊕ 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅)
86		n0 4260	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
87	85, 86	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊕ 𝑣) ⊊ 𝑈 → ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
88	84, 87	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
89		pgpfac1.o	. . . . . . . . . . . . . . . 16 ⊢ 𝑂 = (od‘𝐺)
90		pgpfac1.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (gEx‘𝐺)
91		pgpfac1.z	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0_g‘𝐺)
92		pgpfac1.l	. . . . . . . . . . . . . . . 16 ⊢ ⊕ = (LSSum‘𝐺)
93		pgpfac1.p	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
94	93	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝑃 pGrp 𝐺)
95	15	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝐺 ∈ Abel)
96	1	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝐵 ∈ Fin)
97		pgpfac1.oe	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
98	97	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → (𝑂‘𝐴) = 𝐸)
99	21	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝑈 ∈ (SubGrp‘𝐺))
100	24	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝐴 ∈ 𝑈)
101		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝑣 ∈ (SubGrp‘𝐺))
102		simprl1 1215	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → (𝑆 ∩ 𝑣) = { 0 })
103	84	adantrr 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → (𝑆 ⊕ 𝑣) ⊊ 𝑈)
104	103	pssssd 4025	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → (𝑆 ⊕ 𝑣) ⊆ 𝑈)
105		simprl3 1217	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
106	82	adantrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → (𝑆 ⊕ 𝑣) = 𝑠)
107		psseq1 4015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → ((𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑦))
108	107	notbid 321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ ¬ 𝑠 ⊊ 𝑦))
109	108	imbi2d 344	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦) ↔ ((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦)))
110	109	ralbidv 3162	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦) ↔ ∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦)))
111		psseq1 4015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑦 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
112		eleq2 2878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
113	111, 112	anbi12d 633	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → ((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) ↔ (𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤)))
114		psseq2 4016	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑠 ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑤))
115	114	notbid 321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (¬ 𝑠 ⊊ 𝑦 ↔ ¬ 𝑠 ⊊ 𝑤))
116	113, 115	imbi12d 348	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑤 → (((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
117	116	cbvralvw 3396	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
118	110, 117	syl6bb 290	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
119	106, 118	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
120	105, 119	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → ∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦))
121		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
122		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (._g‘𝐺) = (._g‘𝐺)
123	26, 14, 5, 89, 90, 91, 92, 94, 95, 96, 98, 99, 100, 101, 102, 104, 120, 121, 122	pgpfac1lem4 19193	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
124	123	expr 460	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → (𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
125	124	exlimdv 1934	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → (∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
126	88, 125	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
127	126	3exp2 1351	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) → ((𝑆 ∩ 𝑣) = { 0 } → ((𝑆 ⊕ 𝑣) = 𝑠 → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))))
128	127	impd 414	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) → (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))))
129	128	rexlimdva 3243	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) → (∃𝑣 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))))
130	79, 129	syl5bi 245	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) → (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))))
131	130	impd 414	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) → ((∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
132	73, 131	sylan2b 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}) → ((∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
133	132	rexlimdva 3243	. . . . 5 ⊢ (𝜑 → (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
134	72, 133	syl5 34	. . . 4 ⊢ (𝜑 → ((∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ∧ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
135	71, 134	mpand 694	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
136	65, 135	syld 47	. 2 ⊢ (𝜑 → (𝑆 ⊊ 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
137	91	0subg 18296	. . . . . 6 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
138	17, 137	syl 17	. . . . 5 ⊢ (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
139	138	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
140	91	subg0cl 18279	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
141	29, 140	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆)
142	141	snssd 4702	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ 𝑆)
143	142	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
144		sseqin2 4142	. . . . 5 ⊢ ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145	143, 144	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
146	92	lsmss2 18785	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 ⊕ { 0 }) = 𝑆)
147	29, 138, 142, 146	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑆 ⊕ { 0 }) = 𝑆)
148	147	eqeq1d 2800	. . . . 5 ⊢ (𝜑 → ((𝑆 ⊕ { 0 }) = 𝑈 ↔ 𝑆 = 𝑈))
149	148	biimpar 481	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ⊕ { 0 }) = 𝑈)
150		ineq2 4133	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ∩ 𝑡) = (𝑆 ∩ { 0 }))
151	150	eqeq1d 2800	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152		oveq2 7143	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ { 0 }))
153	152	eqeq1d 2800	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ⊕ 𝑡) = 𝑈 ↔ (𝑆 ⊕ { 0 }) = 𝑈))
154	151, 153	anbi12d 633	. . . . 5 ⊢ (𝑡 = { 0 } → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 ⊕ { 0 }) = 𝑈)))
155	154	rspcev 3571	. . . 4 ⊢ (({ 0 } ∈ (SubGrp‘𝐺) ∧ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 ⊕ { 0 }) = 𝑈)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
156	139, 145, 149, 155	syl12anc 835	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
157	156	ex 416	. 2 ⊢ (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
158	26	mrcsscl 16883	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
159	20, 32, 21, 158	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
160	14, 159	eqsstrid 3963	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑈)
161		sspss 4027	. . 3 ⊢ (𝑆 ⊆ 𝑈 ↔ (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
162	160, 161	sylib 221	. 2 ⊢ (𝜑 → (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
163	136, 157, 162	mpjaod 857	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ⊊ wpss 3882 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 class class class wbr 5030 Or wor 5437 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 [⊊] crpss 7428 Fincfn 8492 cardccrd 9348 Basecbs 16475 0_gc0g 16705 Moorecmre 16845 mrClscmrc 16846 ACScacs 16848 Grpcgrp 18095 ._gcmg 18216 SubGrpcsubg 18265 odcod 18644 gExcgex 18645 pGrp cpgp 18646 LSSumclsm 18751 Abelcabl 18899

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rpss 7429 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-dvds 15600 df-gcd 15834 df-prm 16006 df-pc 16164 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-eqg 18270 df-ga 18412 df-cntz 18439 df-od 18648 df-gex 18649 df-pgp 18650 df-lsm 18753 df-cmn 18900 df-abl 18901

This theorem is referenced by: pgpfac1 19195

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