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Theorem pgpfac1lem5 20010

Description: Lemma for pgpfac1 20011. (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 9219	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
3	1, 2	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 𝐵 ∈ Fin)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝒫 𝐵 ∈ Fin)
5		pgpfac1.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgss 19057	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (SubGrp‘𝐺) → 𝑣 ⊆ 𝐵)
7	6	3ad2ant2 1134	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ⊆ 𝐵)
8		velpw 4559	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝒫 𝐵 ↔ 𝑣 ⊆ 𝐵)
9	7, 8	sylibr 234	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ∈ 𝒫 𝐵)
10	9	rabssdv 4026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ⊆ 𝒫 𝐵)
11	4, 10	ssfid 9169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12		finnum 9860	. . . . . . 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 19714	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Grp)
18	5	subgacs 19090	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
19		acsmre 17575	. . . . . . . . . . . 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 19057	. . . . . . . . . . . . 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 3934	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
26		pgpfac1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
27	26	mrcsncl 17535	. . . . . . . . . . 11 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
28	20, 25, 27	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
29	14, 28	eqeltrid 2840	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
31		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ⊊ 𝑈)
32	24	snssd 4765	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ⊆ 𝑈)
33	32, 23	sstrd 3944	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
34	20, 26, 33	mrcssidd 17548	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
35	34, 14	sseqtrrdi 3975	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
36		snssg 4740	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
37	25, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
38	35, 37	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
39	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝐴 ∈ 𝑆)
40		psseq1 4042	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝑣 ⊊ 𝑈 ↔ 𝑆 ⊊ 𝑈))
41		eleq2 2825	. . . . . . . . . 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 3576	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑆 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑆)) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
44	30, 31, 39, 43	syl12anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
45		rabn0 4341	. . . . . . 7 ⊢ ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
46	44, 45	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅)
47		simpr1 1195	. . . . . . . . 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 1196	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ≠ ∅)
49	11	adantr 480	. . . . . . . . . . 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 9169	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ∈ Fin)
51		simpr3 1197	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → [⊊] Or 𝑢)
52		fin1a2lem10 10319	. . . . . . . . . 10 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ 𝑢)
53	48, 50, 51, 52	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → ∪ 𝑢 ∈ 𝑢)
54	47, 53	sseldd 3934	. . . . . . . 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 412	. . . . . . 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 10415	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤)
59		psseq1 4042	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
60		eleq2 2825	. . . . . . . 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 3652	. . . . . 6 ⊢ (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
63	62	rexbii 3083	. . . . 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 218	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
65	64	ex 412	. . 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 4042	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝑣 ⊊ 𝑈 ↔ 𝑠 ⊊ 𝑈))
68		eleq2 2825	. . . . . . 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 3652	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))
72		r19.29 3099	. . . . 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 3646	. . . . . . 7 ⊢ (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
74		ineq2 4166	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ∩ 𝑡) = (𝑆 ∩ 𝑣))
75	74	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ 𝑣))
77	76	eqeq1d 2738	. . . . . . . . . . 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 3215	. . . . . . . . 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 726	. . . . . . . . . . . . . . 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 1196	. . . . . . . . . . . . . . . 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 4047	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 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 4321	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊕ 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅)
86		n0 4305	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
87	85, 86	sylib 218	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . 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 4052	. . . . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . 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 4042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → ((𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑦))
108	107	notbid 318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ ¬ 𝑠 ⊊ 𝑦))
109	108	imbi2d 340	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → (((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ (𝑆 ⊕ 𝑣) ⊊ 𝑦) ↔ ((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦)))
110	109	ralbidv 3159	. . . . . . . . . . . . . . . . . . 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 4042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑦 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
112		eleq2 2825	. . . . . . . . . . . . . . . . . . . . . 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 4043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑠 ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑤))
115	114	notbid 318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (¬ 𝑠 ⊊ 𝑦 ↔ ¬ 𝑠 ⊊ 𝑤))
116	113, 115	imbi12d 344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑤 → (((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤)))
117	116	cbvralvw 3214	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
118	110, 117	bitrdi 287	. . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 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 20009	. . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . 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 1355	. . . . . . . . . . 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 410	. . . . . . . . . 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 3137	. . . . . . . . 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	biimtrid 242	. . . . . . . 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 410	. . . . . . 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 594	. . . . . 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 3137	. . . . 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 695	. . 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 19081	. . . . . 6 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
138	17, 137	syl 17	. . . . 5 ⊢ (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
139	138	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
140	91	subg0cl 19064	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
141	29, 140	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆)
142	141	snssd 4765	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ 𝑆)
143	142	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
144		sseqin2 4175	. . . . 5 ⊢ ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145	143, 144	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
146	92	lsmss2 19596	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 ⊕ { 0 }) = 𝑆)
147	29, 138, 142, 146	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑆 ⊕ { 0 }) = 𝑆)
148	147	eqeq1d 2738	. . . . 5 ⊢ (𝜑 → ((𝑆 ⊕ { 0 }) = 𝑈 ↔ 𝑆 = 𝑈))
149	148	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ⊕ { 0 }) = 𝑈)
150		ineq2 4166	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ∩ 𝑡) = (𝑆 ∩ { 0 }))
151	150	eqeq1d 2738	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152		oveq2 7366	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ { 0 }))
153	152	eqeq1d 2738	. . . . . 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 3576	. . . 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 836	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))
157	156	ex 412	. 2 ⊢ (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
158	26	mrcsscl 17543	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
159	20, 32, 21, 158	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
160	14, 159	eqsstrid 3972	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑈)
161		sspss 4054	. . 3 ⊢ (𝑆 ⊆ 𝑈 ↔ (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
162	160, 161	sylib 218	. 2 ⊢ (𝜑 → (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
163	136, 157, 162	mpjaod 860	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 ∀wal 1539 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 {crab 3399 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 ⊊ wpss 3902 ∅c0 4285 𝒫 cpw 4554 {csn 4580 ∪ cuni 4863 class class class wbr 5098 Or wor 5531 dom cdm 5624 ‘cfv 6492 (class class class)co 7358 [⊊] crpss 7667 Fincfn 8883 cardccrd 9847 Basecbs 17136 0_gc0g 17359 Moorecmre 17501 mrClscmrc 17502 ACScacs 17504 Grpcgrp 18863 ._gcmg 18997 SubGrpcsubg 19050 odcod 19453 gExcgex 19454 pGrp cpgp 19455 LSSumclsm 19563 Abelcabl 19710

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-xnn0 12475 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-dvds 16180 df-gcd 16422 df-prm 16599 df-pc 16765 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-eqg 19055 df-ga 19219 df-cntz 19246 df-od 19457 df-gex 19458 df-pgp 19459 df-lsm 19565 df-cmn 19711 df-abl 19712

This theorem is referenced by: pgpfac1 20011

W3C validator