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Theorem cdleme26eALTN 36890
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), fz(s), fz(t) respectively. When t v = p q, fz(s) fz(t) v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme26.b 𝐵 = (Base‘𝐾)
cdleme26.l = (le‘𝐾)
cdleme26.j = (join‘𝐾)
cdleme26.m = (meet‘𝐾)
cdleme26.a 𝐴 = (Atoms‘𝐾)
cdleme26.h 𝐻 = (LHyp‘𝐾)
cdleme26eALT.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme26eALT.f 𝐹 = ((𝑦 𝑈) (𝑄 ((𝑃 𝑦) 𝑊)))
cdleme26eALT.g 𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme26eALT.n 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑦) 𝑊)))
cdleme26eALT.o 𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑧) 𝑊)))
cdleme26eALT.i 𝐼 = (𝑢𝐵𝑦𝐴 ((¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) → 𝑢 = 𝑁))
cdleme26eALT.e 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
Assertion
Ref Expression
cdleme26eALTN ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐼 (𝐸 𝑉))
Distinct variable groups:   𝑦,𝑧,𝑢,𝐴   𝑦,𝐵,𝑧,𝑢   𝑦,𝐻,𝑧   𝑦, ,𝑧,𝑢   𝑦,𝐾,𝑧   𝑦, ,𝑧,𝑢   𝑦, ,𝑧,𝑢   𝑢,𝑁   𝑢,𝑂   𝑦,𝑃,𝑧,𝑢   𝑦,𝑄,𝑧,𝑢   𝑦,𝑆,𝑢   𝑧,𝑇,𝑢   𝑦,𝑈,𝑧,𝑢   𝑦,𝑊,𝑧,𝑢
Allowed substitution hints:   𝑆(𝑧)   𝑇(𝑦)   𝐸(𝑦,𝑧,𝑢)   𝐹(𝑦,𝑧,𝑢)   𝐺(𝑦,𝑧,𝑢)   𝐻(𝑢)   𝐼(𝑦,𝑧,𝑢)   𝐾(𝑢)   𝑁(𝑦,𝑧)   𝑂(𝑦,𝑧)   𝑉(𝑦,𝑧,𝑢)

Proof of Theorem cdleme26eALTN
StepHypRef Expression
1 simp11l 1264 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐾 ∈ HL)
2 simp11r 1265 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑊𝐻)
3 simp231 1297 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑇𝐴)
4 simp12 1184 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5 simp13 1185 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
6 simp21 1186 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑃𝑄)
7 simp221 1294 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑆𝐴)
8 simp31 1189 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → (𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)))
9 simp21 1186 . . . . 5 (((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑦𝐴)
1093ad2ant3 1115 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑦𝐴)
11 simp322 1304 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ¬ 𝑦 𝑊)
12 simp31 1189 . . . . . 6 (((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑧𝐴)
13123ad2ant3 1115 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑧𝐴)
14 simp332 1307 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ¬ 𝑧 𝑊)
1513, 14jca 504 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → (𝑧𝐴 ∧ ¬ 𝑧 𝑊))
1610, 11, 15jca31 507 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ((𝑦𝐴 ∧ ¬ 𝑦 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊)))
17 cdleme26.l . . . 4 = (le‘𝐾)
18 cdleme26.j . . . 4 = (join‘𝐾)
19 cdleme26.m . . . 4 = (meet‘𝐾)
20 cdleme26.a . . . 4 𝐴 = (Atoms‘𝐾)
21 cdleme26.h . . . 4 𝐻 = (LHyp‘𝐾)
22 cdleme26eALT.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
23 cdleme26eALT.f . . . 4 𝐹 = ((𝑦 𝑈) (𝑄 ((𝑃 𝑦) 𝑊)))
24 cdleme26eALT.g . . . 4 𝐺 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
25 cdleme26eALT.n . . . 4 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑦) 𝑊)))
26 cdleme26eALT.o . . . 4 𝑂 = ((𝑃 𝑄) (𝐺 ((𝑇 𝑧) 𝑊)))
2717, 18, 19, 20, 21, 22, 23, 24, 25, 26cdleme22eALTN 36874 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝑇𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ (𝑆𝐴 ∧ (𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ ((𝑦𝐴 ∧ ¬ 𝑦 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊)))) → 𝑁 (𝑂 𝑉))
281, 2, 3, 4, 5, 6, 7, 8, 16, 27syl333anc 1382 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑁 (𝑂 𝑉))
29 simp11 1183 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 simp222 1295 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ¬ 𝑆 𝑊)
31 simp223 1296 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
32 cdleme26.b . . . . 5 𝐵 = (Base‘𝐾)
33 cdleme26eALT.i . . . . 5 𝐼 = (𝑢𝐵𝑦𝐴 ((¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) → 𝑢 = 𝑁))
3432, 17, 18, 19, 20, 21, 22, 23, 25, 33cdleme25cl 36886 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑃𝑄𝑆 (𝑃 𝑄))) → 𝐼𝐵)
3529, 4, 5, 7, 30, 6, 31, 34syl322anc 1378 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐼𝐵)
36 simp323 1305 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ¬ 𝑦 (𝑃 𝑄))
3732fvexi 6507 . . . 4 𝐵 ∈ V
3837, 33riotasv 35488 . . 3 ((𝐼𝐵𝑦𝐴 ∧ (¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄))) → 𝐼 = 𝑁)
3935, 10, 11, 36, 38syl112anc 1354 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐼 = 𝑁)
40 simp232 1298 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ¬ 𝑇 𝑊)
41 simp233 1299 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝑇 (𝑃 𝑄))
42 cdleme26eALT.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
4332, 17, 18, 19, 20, 21, 22, 24, 26, 42cdleme25cl 36886 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → 𝐸𝐵)
4429, 4, 5, 3, 40, 6, 41, 43syl322anc 1378 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐸𝐵)
45 simp333 1308 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → ¬ 𝑧 (𝑃 𝑄))
4637, 42riotasv 35488 . . . 4 ((𝐸𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐸 = 𝑂)
4744, 13, 14, 45, 46syl112anc 1354 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐸 = 𝑂)
4847oveq1d 6985 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → (𝐸 𝑉) = (𝑂 𝑉))
4928, 39, 483brtr4d 4955 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊𝑆 (𝑃 𝑄)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊𝑇 (𝑃 𝑄))) ∧ ((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))) → 𝐼 (𝐸 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2048  wne 2961  wral 3082   class class class wbr 4923  cfv 6182  crio 6930  (class class class)co 6970  Basecbs 16329  lecple 16418  joincjn 17402  meetcmee 17403  Atomscatm 35792  HLchlt 35879  LHypclh 36513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-riotaBAD 35482
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-undef 7735  df-proset 17386  df-poset 17404  df-plt 17416  df-lub 17432  df-glb 17433  df-join 17434  df-meet 17435  df-p0 17497  df-p1 17498  df-lat 17504  df-clat 17566  df-oposet 35705  df-ol 35707  df-oml 35708  df-covers 35795  df-ats 35796  df-atl 35827  df-cvlat 35851  df-hlat 35880  df-llines 36027  df-lplanes 36028  df-lvols 36029  df-lines 36030  df-psubsp 36032  df-pmap 36033  df-padd 36325  df-lhyp 36517
This theorem is referenced by: (None)
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