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Theorem pgpfac1lem5 19201

Description: Lemma for pgpfac1 19202. (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 8819	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
3	1, 2	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 𝐵 ∈ Fin)
4	3	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝒫 𝐵 ∈ Fin)
5		pgpfac1.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgss 18280	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (SubGrp‘𝐺) → 𝑣 ⊆ 𝐵)
7	6	3ad2ant2 1130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ⊆ 𝐵)
8		velpw 4544	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝒫 𝐵 ↔ 𝑣 ⊆ 𝐵)
9	7, 8	sylibr 236	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ∈ 𝒫 𝐵)
10	9	rabssdv 4051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ⊆ 𝒫 𝐵)
11	4, 10	ssfid 8741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12		finnum 9377	. . . . . . 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 18911	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Grp)
18	5	subgacs 18313	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
19		acsmre 16923	. . . . . . . . . . . 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 18280	. . . . . . . . . . . . 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 3968	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
26		pgpfac1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
27	26	mrcsncl 16883	. . . . . . . . . . 11 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
28	20, 25, 27	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
29	14, 28	eqeltrid 2917	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
30	29	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
31		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ⊊ 𝑈)
32	24	snssd 4742	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ⊆ 𝑈)
33	32, 23	sstrd 3977	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
34	20, 26, 33	mrcssidd 16896	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
35	34, 14	sseqtrrdi 4018	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
36		snssg 4717	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
37	25, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
38	35, 37	mpbird 259	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
39	38	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝐴 ∈ 𝑆)
40		psseq1 4064	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝑣 ⊊ 𝑈 ↔ 𝑆 ⊊ 𝑈))
41		eleq2 2901	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑆))
42	40, 41	anbi12d 632	. . . . . . . . 9 ⊢ (𝑣 = 𝑆 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑆 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑆)))
43	42	rspcev 3623	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑆 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑆)) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
44	30, 31, 39, 43	syl12anc 834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
45		rabn0 4339	. . . . . . 7 ⊢ ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
46	44, 45	sylibr 236	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅)
47		simpr1 1190	. . . . . . . . 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 1191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ≠ ∅)
49	11	adantr 483	. . . . . . . . . . 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 8741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ∈ Fin)
51		simpr3 1192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → [⊊] Or 𝑢)
52		fin1a2lem10 9831	. . . . . . . . . 10 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ 𝑢)
53	48, 50, 51, 52	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → ∪ 𝑢 ∈ 𝑢)
54	47, 53	sseldd 3968	. . . . . . . 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 415	. . . . . . 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 9927	. . . . . 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 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤)
59		psseq1 4064	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
60		eleq2 2901	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑤))
61	59, 60	anbi12d 632	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤)))
62	61	ralrab 3685	. . . . . 6 ⊢ (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
63	62	rexbii 3247	. . . . 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 220	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
65	64	ex 415	. . 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 4064	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝑣 ⊊ 𝑈 ↔ 𝑠 ⊊ 𝑈))
68		eleq2 2901	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑠))
69	67, 68	anbi12d 632	. . . . . 6 ⊢ (𝑣 = 𝑠 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
70	69	ralrab 3685	. . . . 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 236	. . . 4 ⊢ (𝜑 → ∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))
72		r19.29 3254	. . . . 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 3680	. . . . . . 7 ⊢ (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
74		ineq2 4183	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ∩ 𝑡) = (𝑆 ∩ 𝑣))
75	74	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76		oveq2 7164	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ 𝑣))
77	76	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑣) = 𝑠))
78	75, 77	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑡 = 𝑣 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠)))
79	78	cbvrexvw 3450	. . . . . . . . 9 ⊢ (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ∃𝑣 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠))
80		simprrl 779	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠))) → 𝑠 ⊊ 𝑈)
81	80	ad2antrr 724	. . . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . 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 4069	. . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . 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 4326	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊕ 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅)
86		n0 4310	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
87	85, 86	sylib 220	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . 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 4074	. . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 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 4064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → ((𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑦))
108	107	notbid 320	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ ¬ 𝑠 ⊊ 𝑦))
109	108	imbi2d 343	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦) ↔ ((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦)))
110	109	ralbidv 3197	. . . . . . . . . . . . . . . . . . 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 4064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑦 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
112		eleq2 2901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
113	111, 112	anbi12d 632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → ((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) ↔ (𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤)))
114		psseq2 4065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑠 ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑤))
115	114	notbid 320	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (¬ 𝑠 ⊊ 𝑦 ↔ ¬ 𝑠 ⊊ 𝑤))
116	113, 115	imbi12d 347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑤 → (((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
117	116	cbvralvw 3449	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
118	110, 117	syl6bb 289	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . 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 19200	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . 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 1350	. . . . . . . . . . 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 413	. . . . . . . . . 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 3284	. . . . . . . . 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 244	. . . . . . . 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 413	. . . . . . 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 595	. . . . . 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 3284	. . . . 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 693	. . 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 18304	. . . . . 6 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
138	17, 137	syl 17	. . . . 5 ⊢ (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
139	138	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
140	91	subg0cl 18287	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
141	29, 140	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆)
142	141	snssd 4742	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ 𝑆)
143	142	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
144		sseqin2 4192	. . . . 5 ⊢ ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145	143, 144	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
146	92	lsmss2 18793	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 ⊕ { 0 }) = 𝑆)
147	29, 138, 142, 146	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → (𝑆 ⊕ { 0 }) = 𝑆)
148	147	eqeq1d 2823	. . . . 5 ⊢ (𝜑 → ((𝑆 ⊕ { 0 }) = 𝑈 ↔ 𝑆 = 𝑈))
149	148	biimpar 480	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ⊕ { 0 }) = 𝑈)
150		ineq2 4183	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ∩ 𝑡) = (𝑆 ∩ { 0 }))
151	150	eqeq1d 2823	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152		oveq2 7164	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ { 0 }))
153	152	eqeq1d 2823	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ⊕ 𝑡) = 𝑈 ↔ (𝑆 ⊕ { 0 }) = 𝑈))
154	151, 153	anbi12d 632	. . . . 5 ⊢ (𝑡 = { 0 } → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 ⊕ { 0 }) = 𝑈)))
155	154	rspcev 3623	. . . 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 834	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
157	156	ex 415	. 2 ⊢ (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
158	26	mrcsscl 16891	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
159	20, 32, 21, 158	syl3anc 1367	. . . 4 ⊢ (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
160	14, 159	eqsstrid 4015	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑈)
161		sspss 4076	. . 3 ⊢ (𝑆 ⊆ 𝑈 ↔ (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
162	160, 161	sylib 220	. 2 ⊢ (𝜑 → (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
163	136, 157, 162	mpjaod 856	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ⊊ wpss 3937 ∅c0 4291 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 class class class wbr 5066 Or wor 5473 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 [⊊] crpss 7448 Fincfn 8509 cardccrd 9364 Basecbs 16483 0_gc0g 16713 Moorecmre 16853 mrClscmrc 16854 ACScacs 16856 Grpcgrp 18103 ._gcmg 18224 SubGrpcsubg 18273 odcod 18652 gExcgex 18653 pGrp cpgp 18654 LSSumclsm 18759 Abelcabl 18907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-rpss 7449 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-dvds 15608 df-gcd 15844 df-prm 16016 df-pc 16174 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-eqg 18278 df-ga 18420 df-cntz 18447 df-od 18656 df-gex 18657 df-pgp 18658 df-lsm 18761 df-cmn 18908 df-abl 18909

This theorem is referenced by: pgpfac1 19202

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