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Theorem cdleme26eALTN 38756
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 1285 . . 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 1286 . . 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 1318 . . 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 1205 . . 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 1206 . . 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 1207 . . 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 1315 . . 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 1210 . . 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 1207 . . . . 5 (((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑦𝐴)
1093ad2ant3 1136 . . . 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 1325 . . . 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 1210 . . . . . 6 (((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑧𝐴)
13123ad2ant3 1136 . . . . 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 1328 . . . . 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 513 . . . 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 516 . . 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 38740 . . 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 1403 . 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 1204 . . . 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 1316 . . . 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 1317 . . . 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 38752 . . . 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 1399 . . 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 1326 . . 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 6854 . . . 4 𝐵 ∈ V
3837, 33riotasv 37353 . . 3 ((𝐼𝐵𝑦𝐴 ∧ (¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄))) → 𝐼 = 𝑁)
3935, 10, 11, 36, 38syl112anc 1375 . 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 1319 . . . . 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 1320 . . . . 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 38752 . . . . 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 1399 . . . 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 1329 . . . 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 37353 . . . 4 ((𝐸𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐸 = 𝑂)
4744, 13, 14, 45, 46syl112anc 1375 . . 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 7367 . 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 5136 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2942  wral 3063   class class class wbr 5104  cfv 6494  crio 7307  (class class class)co 7352  Basecbs 17043  lecple 17100  joincjn 18160  meetcmee 18161  Atomscatm 37657  HLchlt 37744  LHypclh 38379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-riotaBAD 37347
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-iin 4956  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7914  df-2nd 7915  df-undef 8197  df-proset 18144  df-poset 18162  df-plt 18179  df-lub 18195  df-glb 18196  df-join 18197  df-meet 18198  df-p0 18274  df-p1 18275  df-lat 18281  df-clat 18348  df-oposet 37570  df-ol 37572  df-oml 37573  df-covers 37660  df-ats 37661  df-atl 37692  df-cvlat 37716  df-hlat 37745  df-llines 37893  df-lplanes 37894  df-lvols 37895  df-lines 37896  df-psubsp 37898  df-pmap 37899  df-padd 38191  df-lhyp 38383
This theorem is referenced by: (None)
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