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Theorem pgpfac1lem5 18406
Description: Lemma for pgpfac1 18407. (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 8212 . . . . . . . . . 10 (𝐵 ∈ Fin ↔ 𝒫 𝐵 ∈ Fin)
31, 2sylib 208 . . . . . . . . 9 (𝜑 → 𝒫 𝐵 ∈ Fin)
43adantr 481 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝒫 𝐵 ∈ Fin)
5 pgpfac1.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
65subgss 17523 . . . . . . . . . . 11 (𝑣 ∈ (SubGrp‘𝐺) → 𝑣𝐵)
763ad2ant2 1081 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑈𝐴𝑣)) → 𝑣𝐵)
8 selpw 4142 . . . . . . . . . 10 (𝑣 ∈ 𝒫 𝐵𝑣𝐵)
97, 8sylibr 224 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ 𝑣 ∈ (SubGrp‘𝐺) ∧ (𝑣𝑈𝐴𝑣)) → 𝑣 ∈ 𝒫 𝐵)
109rabssdv 3666 . . . . . . . 8 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ⊆ 𝒫 𝐵)
11 ssfi 8131 . . . . . . . 8 ((𝒫 𝐵 ∈ Fin ∧ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ⊆ 𝒫 𝐵) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
124, 10, 11syl2anc 692 . . . . . . 7 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
13 finnum 8725 . . . . . . 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 18126 . . . . . . . . . . . . 13 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
1816, 17syl 17 . . . . . . . . . . . 12 (𝜑𝐺 ∈ Grp)
195subgacs 17557 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
20 acsmre 16241 . . . . . . . . . . . 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 17523 . . . . . . . . . . . . 13 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
2422, 23syl 17 . . . . . . . . . . . 12 (𝜑𝑈𝐵)
25 pgpfac1.au . . . . . . . . . . . 12 (𝜑𝐴𝑈)
2624, 25sseldd 3588 . . . . . . . . . . 11 (𝜑𝐴𝐵)
27 pgpfac1.k . . . . . . . . . . . 12 𝐾 = (mrCls‘(SubGrp‘𝐺))
2827mrcsncl 16200 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
2921, 26, 28syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
3015, 29syl5eqel 2702 . . . . . . . . 9 (𝜑𝑆 ∈ (SubGrp‘𝐺))
3130adantr 481 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
32 simpr 477 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝑆𝑈)
3325snssd 4314 . . . . . . . . . . . . 13 (𝜑 → {𝐴} ⊆ 𝑈)
3433, 24sstrd 3597 . . . . . . . . . . . 12 (𝜑 → {𝐴} ⊆ 𝐵)
3521, 27, 34mrcssidd 16213 . . . . . . . . . . 11 (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
3635, 15syl6sseqr 3636 . . . . . . . . . 10 (𝜑 → {𝐴} ⊆ 𝑆)
37 snssg 4301 . . . . . . . . . . 11 (𝐴𝐵 → (𝐴𝑆 ↔ {𝐴} ⊆ 𝑆))
3826, 37syl 17 . . . . . . . . . 10 (𝜑 → (𝐴𝑆 ↔ {𝐴} ⊆ 𝑆))
3936, 38mpbird 247 . . . . . . . . 9 (𝜑𝐴𝑆)
4039adantr 481 . . . . . . . 8 ((𝜑𝑆𝑈) → 𝐴𝑆)
41 psseq1 3677 . . . . . . . . . 10 (𝑣 = 𝑆 → (𝑣𝑈𝑆𝑈))
42 eleq2 2687 . . . . . . . . . 10 (𝑣 = 𝑆 → (𝐴𝑣𝐴𝑆))
4341, 42anbi12d 746 . . . . . . . . 9 (𝑣 = 𝑆 → ((𝑣𝑈𝐴𝑣) ↔ (𝑆𝑈𝐴𝑆)))
4443rspcev 3298 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑆𝑈𝐴𝑆)) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
4531, 32, 40, 44syl12anc 1321 . . . . . . 7 ((𝜑𝑆𝑈) → ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
46 rabn0 3937 . . . . . . 7 ({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ (SubGrp‘𝐺)(𝑣𝑈𝐴𝑣))
4745, 46sylibr 224 . . . . . 6 ((𝜑𝑆𝑈) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ≠ ∅)
48 simpr1 1065 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})
49 simpr2 1066 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ≠ ∅)
5012adantr 481 . . . . . . . . . . 11 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin)
51 ssfi 8131 . . . . . . . . . . 11 (({𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∈ Fin ∧ 𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}) → 𝑢 ∈ Fin)
5250, 48, 51syl2anc 692 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ∈ Fin)
53 simpr3 1067 . . . . . . . . . 10 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → [] Or 𝑢)
54 fin1a2lem10 9182 . . . . . . . . . 10 ((𝑢 ≠ ∅ ∧ 𝑢 ∈ Fin ∧ [] Or 𝑢) → 𝑢𝑢)
5549, 52, 53, 54syl3anc 1323 . . . . . . . . 9 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢𝑢)
5648, 55sseldd 3588 . . . . . . . 8 (((𝜑𝑆𝑈) ∧ (𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢)) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)})
5756ex 450 . . . . . . 7 ((𝜑𝑆𝑈) → ((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}))
5857alrimiv 1852 . . . . . 6 ((𝜑𝑆𝑈) → ∀𝑢((𝑢 ⊆ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ∧ 𝑢 ≠ ∅ ∧ [] Or 𝑢) → 𝑢 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}))
59 zornn0g 9278 . . . . . 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 1323 . . . . 5 ((𝜑𝑆𝑈) → ∃𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)}∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤)
61 psseq1 3677 . . . . . . . 8 (𝑣 = 𝑤 → (𝑣𝑈𝑤𝑈))
62 eleq2 2687 . . . . . . . 8 (𝑣 = 𝑤 → (𝐴𝑣𝐴𝑤))
6361, 62anbi12d 746 . . . . . . 7 (𝑣 = 𝑤 → ((𝑣𝑈𝐴𝑣) ↔ (𝑤𝑈𝐴𝑤)))
6463ralrab 3354 . . . . . 6 (∀𝑤 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ¬ 𝑠𝑤 ↔ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))
6564rexbii 3035 . . . . 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 450 . . 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 3677 . . . . . . 7 (𝑣 = 𝑠 → (𝑣𝑈𝑠𝑈))
70 eleq2 2687 . . . . . . 7 (𝑣 = 𝑠 → (𝐴𝑣𝐴𝑠))
7169, 70anbi12d 746 . . . . . 6 (𝑣 = 𝑠 → ((𝑣𝑈𝐴𝑣) ↔ (𝑠𝑈𝐴𝑠)))
7271ralrab 3354 . . . . 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 3066 . . . . 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 3350 . . . . . . 7 (𝑠 ∈ {𝑣 ∈ (SubGrp‘𝐺) ∣ (𝑣𝑈𝐴𝑣)} ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠)))
76 ineq2 3791 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (𝑆𝑡) = (𝑆𝑣))
7776eqeq1d 2623 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((𝑆𝑡) = { 0 } ↔ (𝑆𝑣) = { 0 }))
78 oveq2 6618 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (𝑆 𝑡) = (𝑆 𝑣))
7978eqeq1d 2623 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((𝑆 𝑡) = 𝑠 ↔ (𝑆 𝑣) = 𝑠))
8077, 79anbi12d 746 . . . . . . . . . 10 (𝑡 = 𝑣 → (((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠)))
8180cbvrexv 3163 . . . . . . . . 9 (∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑠) ↔ ∃𝑣 ∈ (SubGrp‘𝐺)((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠))
82 simprrl 803 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) → 𝑠𝑈)
8382ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → 𝑠𝑈)
84 simpr2 1066 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑠 ∈ (SubGrp‘𝐺) ∧ (𝑠𝑈𝐴𝑠))) ∧ 𝑣 ∈ (SubGrp‘𝐺)) ∧ ((𝑆𝑣) = { 0 } ∧ (𝑆 𝑣) = 𝑠 ∧ ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤))) → (𝑆 𝑣) = 𝑠)
8584psseq1d 3682 . . . . . . . . . . . . . . 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 3924 . . . . . . . . . . . . . . 15 ((𝑆 𝑣) ⊊ 𝑈 → (𝑈 ∖ (𝑆 𝑣)) ≠ ∅)
88 n0 3912 . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . 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 1104 . . . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . . 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 3687 . . . . . . . . . . . . . . . 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 1106 . . . . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . . . 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 3677 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 𝑣) = 𝑠 → ((𝑆 𝑣) ⊊ 𝑦𝑠𝑦))
110109notbid 308 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 𝑣) = 𝑠 → (¬ (𝑆 𝑣) ⊊ 𝑦 ↔ ¬ 𝑠𝑦))
111110imbi2d 330 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 𝑣) = 𝑠 → (((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦)))
112111ralbidv 2981 . . . . . . . . . . . . . . . . . . 19 ((𝑆 𝑣) = 𝑠 → (∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ (𝑆 𝑣) ⊊ 𝑦) ↔ ∀𝑦 ∈ (SubGrp‘𝐺)((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦)))
113 psseq1 3677 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝑦𝑈𝑤𝑈))
114 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝐴𝑦𝐴𝑤))
115113, 114anbi12d 746 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑤 → ((𝑦𝑈𝐴𝑦) ↔ (𝑤𝑈𝐴𝑤)))
116 psseq2 3678 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑤 → (𝑠𝑦𝑠𝑤))
117116notbid 308 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑤 → (¬ 𝑠𝑦 ↔ ¬ 𝑠𝑤))
118115, 117imbi12d 334 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑤 → (((𝑦𝑈𝐴𝑦) → ¬ 𝑠𝑦) ↔ ((𝑤𝑈𝐴𝑤) → ¬ 𝑠𝑤)))
119118cbvralv 3162 . . . . . . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . 16 (.g𝐺) = (.g𝐺)
12527, 15, 5, 91, 92, 93, 94, 96, 97, 98, 100, 101, 102, 103, 104, 106, 122, 123, 124pgpfac1lem4 18405 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 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 1858 . . . . . . . . . . . . 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 1282 . . . . . . . . . . 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 447 . . . . . . . . . 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 3025 . . . . . . . . 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 447 . . . . . . 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 492 . . . . . 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 3025 . . . . 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 710 . . 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 17547 . . . . . 6 (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
14018, 139syl 17 . . . . 5 (𝜑 → { 0 } ∈ (SubGrp‘𝐺))
141140adantr 481 . . . 4 ((𝜑𝑆 = 𝑈) → { 0 } ∈ (SubGrp‘𝐺))
14293subg0cl 17530 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → 0𝑆)
14330, 142syl 17 . . . . . . 7 (𝜑0𝑆)
144143snssd 4314 . . . . . 6 (𝜑 → { 0 } ⊆ 𝑆)
145144adantr 481 . . . . 5 ((𝜑𝑆 = 𝑈) → { 0 } ⊆ 𝑆)
146 sseqin2 3800 . . . . 5 ({ 0 } ⊆ 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
147145, 146sylib 208 . . . 4 ((𝜑𝑆 = 𝑈) → (𝑆 ∩ { 0 }) = { 0 })
14894lsmss2 18009 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ { 0 } ∈ (SubGrp‘𝐺) ∧ { 0 } ⊆ 𝑆) → (𝑆 { 0 }) = 𝑆)
14930, 140, 144, 148syl3anc 1323 . . . . . 6 (𝜑 → (𝑆 { 0 }) = 𝑆)
150149eqeq1d 2623 . . . . 5 (𝜑 → ((𝑆 { 0 }) = 𝑈𝑆 = 𝑈))
151150biimpar 502 . . . 4 ((𝜑𝑆 = 𝑈) → (𝑆 { 0 }) = 𝑈)
152 ineq2 3791 . . . . . . 7 (𝑡 = { 0 } → (𝑆𝑡) = (𝑆 ∩ { 0 }))
153152eqeq1d 2623 . . . . . 6 (𝑡 = { 0 } → ((𝑆𝑡) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
154 oveq2 6618 . . . . . . 7 (𝑡 = { 0 } → (𝑆 𝑡) = (𝑆 { 0 }))
155154eqeq1d 2623 . . . . . 6 (𝑡 = { 0 } → ((𝑆 𝑡) = 𝑈 ↔ (𝑆 { 0 }) = 𝑈))
156153, 155anbi12d 746 . . . . 5 (𝑡 = { 0 } → (((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 { 0 }) = 𝑈)))
157156rspcev 3298 . . . 4 (({ 0 } ∈ (SubGrp‘𝐺) ∧ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 { 0 }) = 𝑈)) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
158141, 147, 151, 157syl12anc 1321 . . 3 ((𝜑𝑆 = 𝑈) → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
159158ex 450 . 2 (𝜑 → (𝑆 = 𝑈 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈)))
16027mrcsscl 16208 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐴} ⊆ 𝑈𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐴}) ⊆ 𝑈)
16121, 33, 22, 160syl3anc 1323 . . . 4 (𝜑 → (𝐾‘{𝐴}) ⊆ 𝑈)
16215, 161syl5eqss 3633 . . 3 (𝜑𝑆𝑈)
163 sspss 3689 . . 3 (𝑆𝑈 ↔ (𝑆𝑈𝑆 = 𝑈))
164162, 163sylib 208 . 2 (𝜑 → (𝑆𝑈𝑆 = 𝑈))
165138, 159, 164mpjaod 396 1 (𝜑 → ∃𝑡 ∈ (SubGrp‘𝐺)((𝑆𝑡) = { 0 } ∧ (𝑆 𝑡) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3556  cin 3558  wss 3559  wpss 3560  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618   Or wor 4999  dom cdm 5079  cfv 5852  (class class class)co 6610   [] crpss 6896  Fincfn 7906  cardccrd 8712  Basecbs 15788  0gc0g 16028  Moorecmre 16170  mrClscmrc 16171  ACScacs 16173  Grpcgrp 17350  .gcmg 17468  SubGrpcsubg 17516  odcod 17872  gExcgex 17873   pGrp cpgp 17874  LSSumclsm 17977  Abelcabl 18122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-rpss 6897  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-er 7694  df-ec 7696  df-qs 7700  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-acn 8719  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-q 11740  df-rp 11784  df-fz 12276  df-fzo 12414  df-fl 12540  df-mod 12616  df-seq 12749  df-exp 12808  df-fac 13008  df-bc 13037  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358  df-dvds 14915  df-gcd 15148  df-prm 15317  df-pc 15473  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-0g 16030  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-grp 17353  df-minusg 17354  df-sbg 17355  df-mulg 17469  df-subg 17519  df-eqg 17521  df-ga 17651  df-cntz 17678  df-od 17876  df-gex 17877  df-pgp 17878  df-lsm 17979  df-cmn 18123  df-abl 18124
This theorem is referenced by:  pgpfac1  18407
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