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Theorem pgpfac1lem5 20099
Description: Lemma for pgpfac1 20100. (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 = (0g𝐺)
pgpfac1.l = (LSSum‘𝐺)
pgpfac1.p (𝜑𝑃 pGrp 𝐺)
pgpfac1.g (𝜑𝐺 ∈ Abel)
pgpfac1.n (𝜑𝐵 ∈ Fin)
pgpfac1.oe (𝜑 → (𝑂𝐴) = 𝐸)
pgpfac1.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
pgpfac1.au (𝜑𝐴𝑈)
pgpfac1.3 (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠𝑈𝐴𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠)))
Assertion
Ref Expression
pgpfac1lem5 (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 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.
StepHypRef Expression
1 pgpfac1.n . . . . . . . . . 10 (𝜑𝐵 ∈ Fin)
2 pwfi 9357 . . . . . . . . . 10 (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
31, 2sylib 218 . . . . . . . . 9 (𝜑 → 𝒫 𝐵 ∈ Fin)
43adantr 480 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝒫 𝐵 ∈ Fin)
5 pgpfac1.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
65subgss 19145 . . . . . . . . . . 11 (𝑣 ∈ (SubGrp‘𝐺) → 𝑣𝐵)
763ad2ant2 1135 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑈𝐴𝑣)) → 𝑣𝐵)
8 velpw 4605 . . . . . . . . . 10 (𝑣 ∈ 𝒫 𝐵𝑣𝐵)
97, 8sylibr 234 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑈𝐴𝑣)) → 𝑣 ∈ 𝒫 𝐵)
109rabssdv 4075 . . . . . . . 8 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ⊆ 𝒫 𝐵)
114, 10ssfid 9301 . . . . . . 7 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
12 finnum 9988 . . . . . . 7 ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ dom card)
1311, 12syl 17 . . . . . 6 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ dom card)
14 pgpfac1.s . . . . . . . . . 10 𝑆 = (𝐾‘{𝐴})
15 pgpfac1.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Abel)
16 ablgrp 19803 . . . . . . . . . . . . 13 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . . . . 12 (𝜑𝐺 ∈ Grp)
185subgacs 19179 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
19 acsmre 17695 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
2017, 18, 193syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
21 pgpfac1.u . . . . . . . . . . . . 13 (𝜑𝑈 ∈ (SubGrp‘𝐺))
225subgss 19145 . . . . . . . . . . . . 13 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
2321, 22syl 17 . . . . . . . . . . . 12 (𝜑𝑈𝐵)
24 pgpfac1.au . . . . . . . . . . . 12 (𝜑𝐴𝑈)
2523, 24sseldd 3984 . . . . . . . . . . 11 (𝜑𝐴𝐵)
26 pgpfac1.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
2726mrcsncl 17655 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
2820, 25, 27syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
2914, 28eqeltrid 2845 . . . . . . . . 9 (𝜑𝑆 ∈ (SubGrp‘𝐺))
3029adantr 480 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
31 simpr 484 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝑆𝑈)
3224snssd 4809 . . . . . . . . . . . . 13 (𝜑 → {𝐴} ⊆ 𝑈)
3332, 23sstrd 3994 . . . . . . . . . . . 12 (𝜑 → {𝐴} ⊆ 𝐵)
3420, 26, 33mrcssidd 17668 . . . . . . . . . . 11 (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
3534, 14sseqtrrdi 4025 . . . . . . . . . 10 (𝜑 → {𝐴} ⊆ 𝑆)
36 snssg 4783 . . . . . . . . . . 11 (𝐴𝐵 → (𝐴𝑆 ↔ {𝐴} ⊆ 𝑆))
3725, 36syl 17 . . . . . . . . . 10 (𝜑 → (𝐴𝑆 ↔ {𝐴} ⊆ 𝑆))
3835, 37mpbird 257 . . . . . . . . 9 (𝜑𝐴𝑆)
3938adantr 480 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝐴𝑆)
40 psseq1 4090 . . . . . . . . . 10 (𝑣 = 𝑆 → (𝑣𝑈𝑆𝑈))
41 eleq2 2830 . . . . . . . . . 10 (𝑣 = 𝑆 → (𝐴𝑣𝐴𝑆))
4240, 41anbi12d 632 . . . . . . . . 9 (𝑣 = 𝑆 → ((𝑣𝑈𝐴𝑣) ↔ (𝑆𝑈𝐴𝑆)))
4342rspcev 3622 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑆𝑈𝐴𝑆)) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
4430, 31, 39, 43syl12anc 837 . . . . . . 7 ((𝜑𝑆𝑈) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
45 rabn0 4389 . . . . . . 7 ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
4644, 45sylibr 234 . . . . . 6 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅)
47 simpr1 1195 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})
48 simpr2 1196 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ≠ ∅)
4911adantr 480 . . . . . . . . . . 11 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
5049, 47ssfid 9301 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ∈ Fin)
51 simpr3 1197 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → [] Or 𝑢)
52 fin1a2lem10 10449 . . . . . . . . . 10 ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [] Or 𝑢) → 𝑢𝑢)
5348, 50, 51, 52syl3anc 1373 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢𝑢)
5447, 53sseldd 3984 . . . . . . . 8 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})
5554ex 412 . . . . . . 7 ((𝜑𝑆𝑈) → ((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}))
5655alrimiv 1927 . . . . . 6 ((𝜑𝑆𝑈) → ∀𝑢((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}))
57 zornn0g 10545 . . . . . 6 (({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ dom card ∧ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅ ∧ ∀𝑢((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤)
5813, 46, 56, 57syl3anc 1373 . . . . 5 ((𝜑𝑆𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤)
59 psseq1 4090 . . . . . . . 8 (𝑣 = 𝑤 → (𝑣𝑈𝑤𝑈))
60 eleq2 2830 . . . . . . . 8 (𝑣 = 𝑤 → (𝐴𝑣𝐴𝑤))
6159, 60anbi12d 632 . . . . . . 7 (𝑣 = 𝑤 → ((𝑣𝑈𝐴𝑣) ↔ (𝑤𝑈𝐴𝑤)))
6261ralrab 3699 . . . . . 6 (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6362rexbii 3094 . . . . 5 (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤 ↔ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6458, 63sylib 218 . . . 4 ((𝜑𝑆𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6564ex 412 . . 3 (𝜑 → (𝑆𝑈 → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
66 pgpfac1.3 . . . . 5 (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠𝑈𝐴𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠)))
67 psseq1 4090 . . . . . . 7 (𝑣 = 𝑠 → (𝑣𝑈𝑠𝑈))
68 eleq2 2830 . . . . . . 7 (𝑣 = 𝑠 → (𝐴𝑣𝐴𝑠))
6967, 68anbi12d 632 . . . . . 6 (𝑣 = 𝑠 → ((𝑣𝑈𝐴𝑣) ↔ (𝑠𝑈𝐴𝑠)))
7069ralrab 3699 . . . . 5 (∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠𝑈𝐴𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠)))
7166, 70sylibr 234 . . . 4 (𝜑 → ∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠))
72 r19.29 3114 . . . . 5 ((∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ∧ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
7369elrab 3692 . . . . . . 7 (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠)))
74 ineq2 4214 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (𝑆𝑡) = (𝑆𝑣))
7574eqeq1d 2739 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((𝑆𝑡) = { 0 } ↔ (𝑆𝑣) = { 0 }))
76 oveq2 7439 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (𝑆 𝑡) = (𝑆 𝑣))
7776eqeq1d 2739 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((𝑆 𝑡) = 𝑠 ↔ (𝑆 𝑣) = 𝑠))
7875, 77anbi12d 632 . . . . . . . . . 10 (𝑡 = 𝑣 → (((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠)))
7978cbvrexvw 3238 . . . . . . . . 9 (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ∃𝑣 ∈ (SubGrp‘𝐺)((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠))
80 simprrl 781 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) → 𝑠𝑈)
8180ad2antrr 726 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → 𝑠𝑈)
82 simpr2 1196 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (𝑆 𝑣) = 𝑠)
8382psseq1d 4095 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → ((𝑆 𝑣) ⊊ 𝑈𝑠𝑈))
8481, 83mpbird 257 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (𝑆 𝑣) ⊊ 𝑈)
85 pssdif 4369 . . . . . . . . . . . . . . 15 ((𝑆 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 𝑣)) ≠ ∅)
86 n0 4353 . . . . . . . . . . . . . . 15 ((𝑈 ∖ (𝑆 𝑣)) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))
8785, 86sylib 218 . . . . . . . . . . . . . 14 ((𝑆 𝑣) ⊊ 𝑈 → ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))
8884, 87syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))
89 pgpfac1.o . . . . . . . . . . . . . . . 16 𝑂 = (od‘𝐺)
90 pgpfac1.e . . . . . . . . . . . . . . . 16 𝐸 = (gEx‘𝐺)
91 pgpfac1.z . . . . . . . . . . . . . . . 16 0 = (0g𝐺)
92 pgpfac1.l . . . . . . . . . . . . . . . 16 = (LSSum‘𝐺)
93 pgpfac1.p . . . . . . . . . . . . . . . . 17 (𝜑𝑃 pGrp 𝐺)
9493ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝑃 pGrp 𝐺)
9515ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝐺 ∈ Abel)
961ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝐵 ∈ Fin)
97 pgpfac1.oe . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑂𝐴) = 𝐸)
9897ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → (𝑂𝐴) = 𝐸)
9921ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝑈 ∈ (SubGrp‘𝐺))
10024ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝐴𝑈)
101 simplr 769 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝑣 ∈ (SubGrp‘𝐺))
102 simprl1 1219 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → (𝑆𝑣) = { 0 })
10384adantrr 717 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → (𝑆 𝑣) ⊊ 𝑈)
104103pssssd 4100 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → (𝑆 𝑣) ⊆ 𝑈)
105 simprl3 1221 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
10682adantrr 717 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → (𝑆 𝑣) = 𝑠)
107 psseq1 4090 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 𝑣) = 𝑠 → ((𝑆 𝑣) ⊊ 𝑦𝑠𝑦))
108107notbid 318 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 𝑣) = 𝑠 → (¬ (𝑆 𝑣) ⊊ 𝑦 ↔ ¬ 𝑠𝑦))
109108imbi2d 340 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 𝑣) = 𝑠 → (((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦)))
110109ralbidv 3178 . . . . . . . . . . . . . . . . . . 19 ((𝑆 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦)))
111 psseq1 4090 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝑦𝑈𝑤𝑈))
112 eleq2 2830 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
113111, 112anbi12d 632 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑤 → ((𝑦𝑈𝐴𝑦) ↔ (𝑤𝑈𝐴𝑤)))
114 psseq2 4091 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝑠𝑦𝑠𝑤))
115114notbid 318 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑤 → (¬ 𝑠𝑦 ↔ ¬ 𝑠𝑤))
116113, 115imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑤 → (((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦) ↔ ((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
117116cbvralvw 3237 . . . . . . . . . . . . . . . . . . 19 (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
118110, 117bitrdi 287 . . . . . . . . . . . . . . . . . 18 ((𝑆 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
119106, 118syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
120105, 119mpbird 257 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → ∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦))
121 simprr 773 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))
122 eqid 2737 . . . . . . . . . . . . . . . 16 (.g𝐺) = (.g𝐺)
12326, 14, 5, 89, 90, 91, 92, 94, 95, 96, 98, 99, 100, 101, 102, 104, 120, 121, 122pgpfac1lem4 20098 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
124123expr 456 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
125124exlimdv 1933 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
12688, 125mpd 15 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
1271263exp2 1355 . . . . . . . . . . 11 (((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) → ((𝑆𝑣) = { 0 } → ((𝑆 𝑣) = 𝑠 → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))))
128127impd 410 . . . . . . . . . 10 (((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) → (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))))
129128rexlimdva 3155 . . . . . . . . 9 ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) → (∃𝑣 ∈ (SubGrp‘𝐺)((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))))
13079, 129biimtrid 242 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) → (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))))
131130impd 410 . . . . . . 7 ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) → ((∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
13273, 131sylan2b 594 . . . . . 6 ((𝜑𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}) → ((∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
133132rexlimdva 3155 . . . . 5 (𝜑 → (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
13472, 133syl5 34 . . . 4 (𝜑 → ((∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ∧ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
13571, 134mpand 695 . . 3 (𝜑 → (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
13665, 135syld 47 . 2 (𝜑 → (𝑆𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
137910subg 19169 . . . . . 6 (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
13817, 137syl 17 . . . . 5 (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
139138adantr 480 . . . 4 ((𝜑𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
14091subg0cl 19152 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → 0𝑆)
14129, 140syl 17 . . . . . . 7 (𝜑0𝑆)
142141snssd 4809 . . . . . 6 (𝜑 → { 0 } ⊆ 𝑆)
143142adantr 480 . . . . 5 ((𝜑𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
144 sseqin2 4223 . . . . 5 ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145143, 144sylib 218 . . . 4 ((𝜑𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
14692lsmss2 19685 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 { 0 }) = 𝑆)
14729, 138, 142, 146syl3anc 1373 . . . . . 6 (𝜑 → (𝑆 { 0 }) = 𝑆)
148147eqeq1d 2739 . . . . 5 (𝜑 → ((𝑆 { 0 }) = 𝑈𝑆 = 𝑈))
149148biimpar 477 . . . 4 ((𝜑𝑆 = 𝑈) → (𝑆 { 0 }) = 𝑈)
150 ineq2 4214 . . . . . . 7 (𝑡 = { 0 } → (𝑆𝑡) = (𝑆 ∩ { 0 }))
151150eqeq1d 2739 . . . . . 6 (𝑡 = { 0 } → ((𝑆𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152 oveq2 7439 . . . . . . 7 (𝑡 = { 0 } → (𝑆 𝑡) = (𝑆 { 0 }))
153152eqeq1d 2739 . . . . . 6 (𝑡 = { 0 } → ((𝑆 𝑡) = 𝑈 ↔ (𝑆 { 0 }) = 𝑈))
154151, 153anbi12d 632 . . . . 5 (𝑡 = { 0 } → (((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 { 0 }) = 𝑈)))
155154rspcev 3622 . . . 4 (({ 0 } ∈ (SubGrp‘𝐺) ∧ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 { 0 }) = 𝑈)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
156139, 145, 149, 155syl12anc 837 . . 3 ((𝜑𝑆 = 𝑈) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
157156ex 412 . 2 (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
15826mrcsscl 17663 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
15920, 32, 21, 158syl3anc 1373 . . . 4 (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
16014, 159eqsstrid 4022 . . 3 (𝜑𝑆𝑈)
161 sspss 4102 . . 3 (𝑆𝑈 ↔ (𝑆𝑈𝑆 = 𝑈))
162160, 161sylib 218 . 2 (𝜑 → (𝑆𝑈𝑆 = 𝑈))
163136, 157, 162mpjaod 861 1 (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cdif 3948  cin 3950  wss 3951  wpss 3952  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143   Or wor 5591  dom cdm 5685  cfv 6561  (class class class)co 7431   [] crpss 7742  Fincfn 8985  cardccrd 9975  Basecbs 17247  0gc0g 17484  Moorecmre 17625  mrClscmrc 17626  ACScacs 17628  Grpcgrp 18951  .gcmg 19085  SubGrpcsubg 19138  odcod 19542  gExcgex 19543   pGrp cpgp 19544  LSSumclsm 19652  Abelcabl 19799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-eqg 19143  df-ga 19308  df-cntz 19335  df-od 19546  df-gex 19547  df-pgp 19548  df-lsm 19654  df-cmn 19800  df-abl 19801
This theorem is referenced by:  pgpfac1  20100
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