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Theorem cdleme26eALTN 35966
 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 1192 . . 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 1193 . . 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 1225 . . 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 1112 . . 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 1113 . . 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 1114 . . 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 1222 . . 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 1117 . . 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 1114 . . . . 5 (((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑦𝐴)
1093ad2ant3 1104 . . . 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 1232 . . . 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 1117 . . . . . 6 (((𝑉𝐴𝑉 𝑊 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑦𝐴 ∧ ¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑧𝐴)
13123ad2ant3 1104 . . . . 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 1235 . . . . 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 553 . . . 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 556 . . 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 35950 . . 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 1398 . 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 1111 . . . 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 1223 . . . 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 1224 . . . 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 35962 . . . 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 1394 . . 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 1233 . . 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))
37 fvex 6239 . . . . 5 (Base‘𝐾) ∈ V
3832, 37eqeltri 2726 . . . 4 𝐵 ∈ V
3938, 33riotasv 34563 . . 3 ((𝐼𝐵𝑦𝐴 ∧ (¬ 𝑦 𝑊 ∧ ¬ 𝑦 (𝑃 𝑄))) → 𝐼 = 𝑁)
4035, 10, 11, 36, 39syl112anc 1370 . 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)
41 simp232 1226 . . . . 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)
42 simp233 1227 . . . . 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))
43 cdleme26eALT.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
4432, 17, 18, 19, 20, 21, 22, 24, 26, 43cdleme25cl 35962 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → 𝐸𝐵)
4529, 4, 5, 3, 41, 6, 42, 44syl322anc 1394 . . . 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)
46 simp333 1236 . . . 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))
4738, 43riotasv 34563 . . . 4 ((𝐸𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐸 = 𝑂)
4845, 13, 14, 46, 47syl112anc 1370 . . 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)
4948oveq1d 6705 . 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))
5028, 40, 493brtr4d 4717 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   class class class wbr 4685  ‘cfv 5926  ℩crio 6650  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  HLchlt 34955  LHypclh 35588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-riotaBAD 34557 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-undef 7444  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592 This theorem is referenced by: (None)
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