pgpfac1lem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pgpfac1lem5		Structured version Visualization version GIF version

Theorem pgpfac1lem5 19951

Description: Lemma for pgpfac1 19952. (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 9180	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
3	1, 2	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 𝐵 ∈ Fin)
4	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝒫 𝐵 ∈ Fin)
5		pgpfac1.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgss 19009	. . . . . . . . . . 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 4607	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝒫 𝐵 ↔ 𝑣 ⊆ 𝐵)
9	7, 8	sylibr 233	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)) → 𝑣 ∈ 𝒫 𝐵)
10	9	rabssdv 4072	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ⊆ 𝒫 𝐵)
11	4, 10	ssfid 9269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12		finnum 9945	. . . . . . 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 19655	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Grp)
18	5	subgacs 19043	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
19		acsmre 17598	. . . . . . . . . . . 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 19009	. . . . . . . . . . . . 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 3983	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
26		pgpfac1.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
27	26	mrcsncl 17558	. . . . . . . . . . 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 2837	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
30	29	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
31		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝑆 ⊊ 𝑈)
32	24	snssd 4812	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ⊆ 𝑈)
33	32, 23	sstrd 3992	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
34	20, 26, 33	mrcssidd 17571	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
35	34, 14	sseqtrrdi 4033	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
36		snssg 4787	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
37	25, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
38	35, 37	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
39	38	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → 𝐴 ∈ 𝑆)
40		psseq1 4087	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝑣 ⊊ 𝑈 ↔ 𝑆 ⊊ 𝑈))
41		eleq2 2822	. . . . . . . . . 10 ⊢ (𝑣 = 𝑆 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑆))
42	40, 41	anbi12d 631	. . . . . . . . 9 ⊢ (𝑣 = 𝑆 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑆 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑆)))
43	42	rspcev 3612	. . . . . . . 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 4385	. . . . . . 7 ⊢ ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣))
46	44, 45	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ≠ ∅)
47		simpr1 1194	. . . . . . . . 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 1195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ≠ ∅)
49	11	adantr 481	. . . . . . . . . . 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 9269	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → 𝑢 ∈ Fin)
51		simpr3 1196	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → [⊊] Or 𝑢)
52		fin1a2lem10 10406	. . . . . . . . . 10 ⊢ ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [⊊] Or 𝑢) → ∪ 𝑢 ∈ 𝑢)
53	48, 50, 51, 52	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊊ 𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ∧ 𝑢 ≠ ∅ ∧ [⊊] Or 𝑢)) → ∪ 𝑢 ∈ 𝑢)
54	47, 53	sseldd 3983	. . . . . . . 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 413	. . . . . . 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 1930	. . . . . 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 10502	. . . . . 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 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤)
59		psseq1 4087	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑣 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
60		eleq2 2822	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑤))
61	59, 60	anbi12d 631	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤)))
62	61	ralrab 3689	. . . . . 6 ⊢ (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ¬ 𝑠 ⊊ 𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
63	62	rexbii 3094	. . . . 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 217	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ⊊ 𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
65	64	ex 413	. . 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 4087	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝑣 ⊊ 𝑈 ↔ 𝑠 ⊊ 𝑈))
68		eleq2 2822	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑠))
69	67, 68	anbi12d 631	. . . . . 6 ⊢ (𝑣 = 𝑠 → ((𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣) ↔ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
70	69	ralrab 3689	. . . . 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 233	. . . 4 ⊢ (𝜑 → ∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠))
72		r19.29 3114	. . . . 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 3683	. . . . . . 7 ⊢ (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑠)))
74		ineq2 4206	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ∩ 𝑡) = (𝑆 ∩ 𝑣))
75	74	eqeq1d 2734	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑣 → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ 𝑣))
77	76	eqeq1d 2734	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑣 → ((𝑆 ⊕ 𝑡) = 𝑠 ↔ (𝑆 ⊕ 𝑣) = 𝑠))
78	75, 77	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑡 = 𝑣 → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑠) ↔ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 ⊕ 𝑣) = 𝑠)))
79	78	cbvrexvw 3235	. . . . . . . . 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 1195	. . . . . . . . . . . . . . . 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 4092	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 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 4366	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊕ 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅)
86		n0 4346	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∖ (𝑆 ⊕ 𝑣)) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑣)))
87	85, 86	sylib 217	. . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . . . 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 4097	. . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . 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 4087	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ⊕ 𝑣) = 𝑠 → ((𝑆 ⊕ 𝑣) ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑦))
108	107	notbid 317	. . . . . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . . 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 4087	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑦 ⊊ 𝑈 ↔ 𝑤 ⊊ 𝑈))
112		eleq2 2822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
113	111, 112	anbi12d 631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → ((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) ↔ (𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤)))
114		psseq2 4088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑤 → (𝑠 ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑤))
115	114	notbid 317	. . . . . . . . . . . . . . . . . . . . 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 3234	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑦) → ¬ 𝑠 ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ 𝑠 ⊊ 𝑤))
118	110, 117	bitrdi 286	. . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 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 19950	. . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . 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 1936	. . . . . . . . . . . . 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 1354	. . . . . . . . . . 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 411	. . . . . . . . . 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 3155	. . . . . . . . 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 241	. . . . . . . 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 411	. . . . . . 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 3155	. . . . 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 19033	. . . . . 6 ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
138	17, 137	syl 17	. . . . 5 ⊢ (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
139	138	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
140	91	subg0cl 19016	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)
141	29, 140	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆)
142	141	snssd 4812	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ 𝑆)
143	142	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
144		sseqin2 4215	. . . . 5 ⊢ ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145	143, 144	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
146	92	lsmss2 19537	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 ⊕ { 0 }) = 𝑆)
147	29, 138, 142, 146	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝑆 ⊕ { 0 }) = 𝑆)
148	147	eqeq1d 2734	. . . . 5 ⊢ (𝜑 → ((𝑆 ⊕ { 0 }) = 𝑈 ↔ 𝑆 = 𝑈))
149	148	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ 𝑆 = 𝑈) → (𝑆 ⊕ { 0 }) = 𝑈)
150		ineq2 4206	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ∩ 𝑡) = (𝑆 ∩ { 0 }))
151	150	eqeq1d 2734	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ∩ 𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152		oveq2 7419	. . . . . . 7 ⊢ (𝑡 = { 0 } → (𝑆 ⊕ 𝑡) = (𝑆 ⊕ { 0 }))
153	152	eqeq1d 2734	. . . . . 6 ⊢ (𝑡 = { 0 } → ((𝑆 ⊕ 𝑡) = 𝑈 ↔ (𝑆 ⊕ { 0 }) = 𝑈))
154	151, 153	anbi12d 631	. . . . 5 ⊢ (𝑡 = { 0 } → (((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 ⊕ { 0 }) = 𝑈)))
155	154	rspcev 3612	. . . 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 413	. 2 ⊢ (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈)))
158	26	mrcsscl 17566	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
159	20, 32, 21, 158	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
160	14, 159	eqsstrid 4030	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑈)
161		sspss 4099	. . 3 ⊢ (𝑆 ⊆ 𝑈 ↔ (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
162	160, 161	sylib 217	. 2 ⊢ (𝜑 → (𝑆 ⊊ 𝑈 ∨ 𝑆 = 𝑈))
163	136, 157, 162	mpjaod 858	1 ⊢ (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆 ∩ 𝑡) = { 0 } ∧ (𝑆 ⊕ 𝑡) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ⊊ wpss 3949 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 class class class wbr 5148 Or wor 5587 dom cdm 5676 ‘cfv 6543 (class class class)co 7411 [⊊] crpss 7714 Fincfn 8941 cardccrd 9932 Basecbs 17146 0_gc0g 17387 Moorecmre 17528 mrClscmrc 17529 ACScacs 17531 Grpcgrp 18821 ._gcmg 18952 SubGrpcsubg 19002 odcod 19394 gExcgex 19395 pGrp cpgp 19396 LSSumclsm 19504 Abelcabl 19651

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-rpss 7715 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-xnn0 12547 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-dvds 16200 df-gcd 16438 df-prm 16611 df-pc 16772 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-0g 17389 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-eqg 19007 df-ga 19156 df-cntz 19183 df-od 19398 df-gex 19399 df-pgp 19400 df-lsm 19506 df-cmn 19652 df-abl 19653

This theorem is referenced by: pgpfac1 19952

W3C validator