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Theorem pgpfac1lem5 18686
Description: Lemma for pgpfac1 18687. (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 8417 . . . . . . . . . 10 (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
31, 2sylib 208 . . . . . . . . 9 (𝜑 → 𝒫 𝐵 ∈ Fin)
43adantr 466 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝒫 𝐵 ∈ Fin)
5 pgpfac1.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
65subgss 17803 . . . . . . . . . . 11 (𝑣 ∈ (SubGrp‘𝐺) → 𝑣𝐵)
763ad2ant2 1128 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑈𝐴𝑣)) → 𝑣𝐵)
8 selpw 4304 . . . . . . . . . 10 (𝑣 ∈ 𝒫 𝐵𝑣𝐵)
97, 8sylibr 224 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑈𝐴𝑣)) → 𝑣 ∈ 𝒫 𝐵)
109rabssdv 3831 . . . . . . . 8 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ⊆ 𝒫 𝐵)
11 ssfi 8336 . . . . . . . 8 ((𝒫 𝐵 ∈ Fin ∧ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ⊆ 𝒫 𝐵) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
124, 10, 11syl2anc 573 . . . . . . 7 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
13 finnum 8974 . . . . . . 7 ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ dom card)
1412, 13syl 17 . . . . . 6 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ dom card)
15 pgpfac1.s . . . . . . . . . 10 𝑆 = (𝐾‘{𝐴})
16 pgpfac1.g . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Abel)
17 ablgrp 18405 . . . . . . . . . . . . 13 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1816, 17syl 17 . . . . . . . . . . . 12 (𝜑𝐺 ∈ Grp)
195subgacs 17837 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
20 acsmre 16520 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
2118, 19, 203syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
22 pgpfac1.u . . . . . . . . . . . . 13 (𝜑𝑈 ∈ (SubGrp‘𝐺))
235subgss 17803 . . . . . . . . . . . . 13 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
2422, 23syl 17 . . . . . . . . . . . 12 (𝜑𝑈𝐵)
25 pgpfac1.au . . . . . . . . . . . 12 (𝜑𝐴𝑈)
2624, 25sseldd 3753 . . . . . . . . . . 11 (𝜑𝐴𝐵)
27 pgpfac1.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
2827mrcsncl 16480 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
2921, 26, 28syl2anc 573 . . . . . . . . . 10 (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
3015, 29syl5eqel 2854 . . . . . . . . 9 (𝜑𝑆 ∈ (SubGrp‘𝐺))
3130adantr 466 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
32 simpr 471 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝑆𝑈)
3325snssd 4475 . . . . . . . . . . . . 13 (𝜑 → {𝐴} ⊆ 𝑈)
3433, 24sstrd 3762 . . . . . . . . . . . 12 (𝜑 → {𝐴} ⊆ 𝐵)
3521, 27, 34mrcssidd 16493 . . . . . . . . . . 11 (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
3635, 15syl6sseqr 3801 . . . . . . . . . 10 (𝜑 → {𝐴} ⊆ 𝑆)
37 snssg 4450 . . . . . . . . . . 11 (𝐴𝐵 → (𝐴𝑆 ↔ {𝐴} ⊆ 𝑆))
3826, 37syl 17 . . . . . . . . . 10 (𝜑 → (𝐴𝑆 ↔ {𝐴} ⊆ 𝑆))
3936, 38mpbird 247 . . . . . . . . 9 (𝜑𝐴𝑆)
4039adantr 466 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝐴𝑆)
41 psseq1 3844 . . . . . . . . . 10 (𝑣 = 𝑆 → (𝑣𝑈𝑆𝑈))
42 eleq2 2839 . . . . . . . . . 10 (𝑣 = 𝑆 → (𝐴𝑣𝐴𝑆))
4341, 42anbi12d 616 . . . . . . . . 9 (𝑣 = 𝑆 → ((𝑣𝑈𝐴𝑣) ↔ (𝑆𝑈𝐴𝑆)))
4443rspcev 3460 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑆𝑈𝐴𝑆)) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
4531, 32, 40, 44syl12anc 1474 . . . . . . 7 ((𝜑𝑆𝑈) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
46 rabn0 4104 . . . . . . 7 ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
4745, 46sylibr 224 . . . . . 6 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅)
48 simpr1 1233 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})
49 simpr2 1235 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ≠ ∅)
5012adantr 466 . . . . . . . . . . 11 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
51 ssfi 8336 . . . . . . . . . . 11 (({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin ∧ 𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}) → 𝑢 ∈ Fin)
5250, 48, 51syl2anc 573 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ∈ Fin)
53 simpr3 1237 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → [] Or 𝑢)
54 fin1a2lem10 9433 . . . . . . . . . 10 ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [] Or 𝑢) → 𝑢𝑢)
5549, 52, 53, 54syl3anc 1476 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢𝑢)
5648, 55sseldd 3753 . . . . . . . 8 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})
5756ex 397 . . . . . . 7 ((𝜑𝑆𝑈) → ((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}))
5857alrimiv 2007 . . . . . 6 ((𝜑𝑆𝑈) → ∀𝑢((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}))
59 zornn0g 9529 . . . . . 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)
6014, 47, 58, 59syl3anc 1476 . . . . 5 ((𝜑𝑆𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤)
61 psseq1 3844 . . . . . . . 8 (𝑣 = 𝑤 → (𝑣𝑈𝑤𝑈))
62 eleq2 2839 . . . . . . . 8 (𝑣 = 𝑤 → (𝐴𝑣𝐴𝑤))
6361, 62anbi12d 616 . . . . . . 7 (𝑣 = 𝑤 → ((𝑣𝑈𝐴𝑣) ↔ (𝑤𝑈𝐴𝑤)))
6463ralrab 3520 . . . . . 6 (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6564rexbii 3189 . . . . 5 (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤 ↔ ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6660, 65sylib 208 . . . 4 ((𝜑𝑆𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6766ex 397 . . 3 (𝜑 → (𝑆𝑈 → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
68 pgpfac1.3 . . . . 5 (𝜑 → ∀𝑠 ∈ (SubGrp‘𝐺)((𝑠𝑈𝐴𝑠) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠)))
69 psseq1 3844 . . . . . . 7 (𝑣 = 𝑠 → (𝑣𝑈𝑠𝑈))
70 eleq2 2839 . . . . . . 7 (𝑣 = 𝑠 → (𝐴𝑣𝐴𝑠))
7169, 70anbi12d 616 . . . . . 6 (𝑣 = 𝑠 → ((𝑣𝑈𝐴𝑣) ↔ (𝑠𝑈𝐴𝑠)))
7271ralrab 3520 . . . . 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)))
7368, 72sylibr 224 . . . 4 (𝜑 → ∀𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠))
74 r19.29 3220 . . . . 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)))
7571elrab 3515 . . . . . . 7 (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠)))
76 ineq2 3959 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (𝑆𝑡) = (𝑆𝑣))
7776eqeq1d 2773 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((𝑆𝑡) = { 0 } ↔ (𝑆𝑣) = { 0 }))
78 oveq2 6801 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (𝑆 𝑡) = (𝑆 𝑣))
7978eqeq1d 2773 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((𝑆 𝑡) = 𝑠 ↔ (𝑆 𝑣) = 𝑠))
8077, 79anbi12d 616 . . . . . . . . . 10 (𝑡 = 𝑣 → (((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠)))
8180cbvrexv 3321 . . . . . . . . 9 (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ∃𝑣 ∈ (SubGrp‘𝐺)((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠))
82 simprrl 766 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) → 𝑠𝑈)
8382ad2antrr 705 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → 𝑠𝑈)
84 simpr2 1235 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (𝑆 𝑣) = 𝑠)
8584psseq1d 3849 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → ((𝑆 𝑣) ⊊ 𝑈𝑠𝑈))
8683, 85mpbird 247 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (𝑆 𝑣) ⊊ 𝑈)
87 pssdif 4092 . . . . . . . . . . . . . . 15 ((𝑆 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 𝑣)) ≠ ∅)
88 n0 4078 . . . . . . . . . . . . . . 15 ((𝑈 ∖ (𝑆 𝑣)) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))
8987, 88sylib 208 . . . . . . . . . . . . . 14 ((𝑆 𝑣) ⊊ 𝑈 → ∃𝑏 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))
9086, 89syl 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)))
91 pgpfac1.o . . . . . . . . . . . . . . . 16 𝑂 = (od‘𝐺)
92 pgpfac1.e . . . . . . . . . . . . . . . 16 𝐸 = (gEx‘𝐺)
93 pgpfac1.z . . . . . . . . . . . . . . . 16 0 = (0g𝐺)
94 pgpfac1.l . . . . . . . . . . . . . . . 16 = (LSSum‘𝐺)
95 pgpfac1.p . . . . . . . . . . . . . . . . 17 (𝜑𝑃 pGrp 𝐺)
9695ad3antrrr 709 . . . . . . . . . . . . . . . 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)
9716ad3antrrr 709 . . . . . . . . . . . . . . . 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)
981ad3antrrr 709 . . . . . . . . . . . . . . . 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)
99 pgpfac1.oe . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑂𝐴) = 𝐸)
10099ad3antrrr 709 . . . . . . . . . . . . . . . 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)
10122ad3antrrr 709 . . . . . . . . . . . . . . . 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))
10225ad3antrrr 709 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ (((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)) ∧ 𝑏 ∈ (𝑈 ∖ (𝑆 𝑣)))) → 𝐴𝑈)
103 simplr 752 . . . . . . . . . . . . . . . 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))
104 simprl1 1266 . . . . . . . . . . . . . . . 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 })
10586adantrr 696 . . . . . . . . . . . . . . . . 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)
106105pssssd 3854 . . . . . . . . . . . . . . . 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)
107 simprl3 1270 . . . . . . . . . . . . . . . . 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))
10884adantrr 696 . . . . . . . . . . . . . . . . . 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)
109 psseq1 3844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 𝑣) = 𝑠 → ((𝑆 𝑣) ⊊ 𝑦𝑠𝑦))
110109notbid 307 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 𝑣) = 𝑠 → (¬ (𝑆 𝑣) ⊊ 𝑦 ↔ ¬ 𝑠𝑦))
111110imbi2d 329 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 𝑣) = 𝑠 → (((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦)))
112111ralbidv 3135 . . . . . . . . . . . . . . . . . . 19 ((𝑆 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦)))
113 psseq1 3844 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝑦𝑈𝑤𝑈))
114 eleq2 2839 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
115113, 114anbi12d 616 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑤 → ((𝑦𝑈𝐴𝑦) ↔ (𝑤𝑈𝐴𝑤)))
116 psseq2 3845 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝑠𝑦𝑠𝑤))
117116notbid 307 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑤 → (¬ 𝑠𝑦 ↔ ¬ 𝑠𝑤))
118115, 117imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑤 → (((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦) ↔ ((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
119118cbvralv 3320 . . . . . . . . . . . . . . . . . . 19 (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
120112, 119syl6bb 276 . . . . . . . . . . . . . . . . . 18 ((𝑆 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
121108, 120syl 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)))
122107, 121mpbird 247 . . . . . . . . . . . . . . . 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))
123 simprr 756 . . . . . . . . . . . . . . . 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)))
124 eqid 2771 . . . . . . . . . . . . . . . 16 (.g𝐺) = (.g𝐺)
12527, 15, 5, 91, 92, 93, 94, 96, 97, 98, 100, 101, 102, 103, 104, 106, 122, 123, 124pgpfac1lem4 18685 . . . . . . . . . . . . . . 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))
126125expr 444 . . . . . . . . . . . . . 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)))
127126exlimdv 2013 . . . . . . . . . . . . 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)))
12890, 127mpd 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))
1291283exp2 1447 . . . . . . . . . . 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)))))
130129impd 396 . . . . . . . . . 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))))
131130rexlimdva 3179 . . . . . . . . 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))))
13281, 131syl5bi 232 . . . . . . . 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))))
133132impd 396 . . . . . . 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)))
13475, 133sylan2b 581 . . . . . 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)))
135134rexlimdva 3179 . . . . 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)))
13674, 135syl5 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)))
13773, 136mpand 675 . . 3 (𝜑 → (∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
13867, 137syld 47 . 2 (𝜑 → (𝑆𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
139930subg 17827 . . . . . 6 (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
14018, 139syl 17 . . . . 5 (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
141140adantr 466 . . . 4 ((𝜑𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
14293subg0cl 17810 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → 0𝑆)
14330, 142syl 17 . . . . . . 7 (𝜑0𝑆)
144143snssd 4475 . . . . . 6 (𝜑 → { 0 } ⊆ 𝑆)
145144adantr 466 . . . . 5 ((𝜑𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
146 sseqin2 3968 . . . . 5 ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
147145, 146sylib 208 . . . 4 ((𝜑𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
14894lsmss2 18288 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 { 0 }) = 𝑆)
14930, 140, 144, 148syl3anc 1476 . . . . . 6 (𝜑 → (𝑆 { 0 }) = 𝑆)
150149eqeq1d 2773 . . . . 5 (𝜑 → ((𝑆 { 0 }) = 𝑈𝑆 = 𝑈))
151150biimpar 463 . . . 4 ((𝜑𝑆 = 𝑈) → (𝑆 { 0 }) = 𝑈)
152 ineq2 3959 . . . . . . 7 (𝑡 = { 0 } → (𝑆𝑡) = (𝑆 ∩ { 0 }))
153152eqeq1d 2773 . . . . . 6 (𝑡 = { 0 } → ((𝑆𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
154 oveq2 6801 . . . . . . 7 (𝑡 = { 0 } → (𝑆 𝑡) = (𝑆 { 0 }))
155154eqeq1d 2773 . . . . . 6 (𝑡 = { 0 } → ((𝑆 𝑡) = 𝑈 ↔ (𝑆 { 0 }) = 𝑈))
156153, 155anbi12d 616 . . . . 5 (𝑡 = { 0 } → (((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 { 0 }) = 𝑈)))
157156rspcev 3460 . . . 4 (({ 0 } ∈ (SubGrp‘𝐺) ∧ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 { 0 }) = 𝑈)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
158141, 147, 151, 157syl12anc 1474 . . 3 ((𝜑𝑆 = 𝑈) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
159158ex 397 . 2 (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
16027mrcsscl 16488 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
16121, 33, 22, 160syl3anc 1476 . . . 4 (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
16215, 161syl5eqss 3798 . . 3 (𝜑𝑆𝑈)
163 sspss 3856 . . 3 (𝑆𝑈 ↔ (𝑆𝑈𝑆 = 𝑈))
164162, 163sylib 208 . 2 (𝜑 → (𝑆𝑈𝑆 = 𝑈))
165138, 159, 164mpjaod 849 1 (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  cdif 3720  cin 3722  wss 3723  wpss 3724  c0 4063  𝒫 cpw 4297  {csn 4316   cuni 4574   class class class wbr 4786   Or wor 5169  dom cdm 5249  cfv 6031  (class class class)co 6793   [] crpss 7083  Fincfn 8109  cardccrd 8961  Basecbs 16064  0gc0g 16308  Moorecmre 16450  mrClscmrc 16451  ACScacs 16453  Grpcgrp 17630  .gcmg 17748  SubGrpcsubg 17796  odcod 18151  gExcgex 18152   pGrp cpgp 18153  LSSumclsm 18256  Abelcabl 18401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-rpss 7084  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-er 7896  df-ec 7898  df-qs 7902  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-acn 8968  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-dvds 15190  df-gcd 15425  df-prm 15593  df-pc 15749  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-mulg 17749  df-subg 17799  df-eqg 17801  df-ga 17930  df-cntz 17957  df-od 18155  df-gex 18156  df-pgp 18157  df-lsm 18258  df-cmn 18402  df-abl 18403
This theorem is referenced by:  pgpfac1  18687
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