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Theorem cdleme26e 36161
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, 2-Feb-2013.)
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)
cdleme26e.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme26e.f 𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme26e.n 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))
cdleme26e.o 𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))
cdleme26e.i 𝐼 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
cdleme26e.e 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
Assertion
Ref Expression
cdleme26e ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐼 (𝐸 𝑉))
Distinct variable groups:   𝑧,𝑢,𝐴   𝑧,𝐵,𝑢   𝑧,𝐻   𝑧, ,𝑢   𝑧,𝐾   𝑧, ,𝑢   𝑧, ,𝑢   𝑢,𝑁   𝑢,𝑂   𝑧,𝑃,𝑢   𝑧,𝑄,𝑢   𝑧,𝑆,𝑢   𝑧,𝑇,𝑢   𝑧,𝑈,𝑢   𝑧,𝑊,𝑢
Allowed substitution hints:   𝐸(𝑧,𝑢)   𝐹(𝑧,𝑢)   𝐻(𝑢)   𝐼(𝑧,𝑢)   𝐾(𝑢)   𝑁(𝑧)   𝑂(𝑧)   𝑉(𝑧,𝑢)

Proof of Theorem cdleme26e
StepHypRef Expression
1 simp11 1244 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp12 1245 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp13 1246 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp21l 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑆𝐴)
5 simp22l 1375 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑇𝐴)
64, 5jca 495 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑆𝐴𝑇𝐴))
7 simp23 1249 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑉𝐴𝑉 𝑊))
8 simp311 1403 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑃𝑄)
9 simp32l 1381 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑇 𝑉) = (𝑃 𝑄))
108, 9jca 495 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)))
11 simp33 1252 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝑧𝐴 ∧ ¬ 𝑧 𝑊))
12 cdleme26.l . . . 4 = (le‘𝐾)
13 cdleme26.j . . . 4 = (join‘𝐾)
14 cdleme26.m . . . 4 = (meet‘𝐾)
15 cdleme26.a . . . 4 𝐴 = (Atoms‘𝐾)
16 cdleme26.h . . . 4 𝐻 = (LHyp‘𝐾)
17 cdleme26e.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
18 cdleme26e.f . . . 4 𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
19 cdleme26e.n . . . 4 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))
20 cdleme26e.o . . . 4 𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))
2112, 13, 14, 15, 16, 17, 18, 19, 20cdleme22e 36146 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴𝑇𝐴)) ∧ ((𝑉𝐴𝑉 𝑊) ∧ (𝑃𝑄 ∧ (𝑇 𝑉) = (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑁 (𝑂 𝑉))
221, 2, 3, 6, 7, 10, 11, 21syl133anc 1498 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑁 (𝑂 𝑉))
23 simp21r 1374 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ¬ 𝑆 𝑊)
24 simp312 1404 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑆 (𝑃 𝑄))
25 cdleme26.b . . . . 5 𝐵 = (Base‘𝐾)
26 cdleme26e.i . . . . 5 𝐼 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
2725, 12, 13, 14, 15, 16, 17, 18, 19, 26cdleme25cl 36159 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑃𝑄𝑆 (𝑃 𝑄))) → 𝐼𝐵)
281, 2, 3, 4, 23, 8, 24, 27syl322anc 1503 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐼𝐵)
29 simp33l 1383 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑧𝐴)
30 simp33r 1384 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ¬ 𝑧 𝑊)
31 simp32r 1382 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ¬ 𝑧 (𝑃 𝑄))
3230, 31jca 495 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))
33 fvex 6342 . . . . 5 (Base‘𝐾) ∈ V
3425, 33eqeltri 2845 . . . 4 𝐵 ∈ V
3534, 26riotasv 34760 . . 3 ((𝐼𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐼 = 𝑁)
3628, 29, 32, 35syl3anc 1475 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐼 = 𝑁)
37 simp22r 1376 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → ¬ 𝑇 𝑊)
38 simp313 1405 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝑇 (𝑃 𝑄))
39 cdleme26e.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
4025, 12, 13, 14, 15, 16, 17, 18, 20, 39cdleme25cl 36159 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑇 (𝑃 𝑄))) → 𝐸𝐵)
411, 2, 3, 5, 37, 8, 38, 40syl322anc 1503 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐸𝐵)
4234, 39riotasv 34760 . . . 4 ((𝐸𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐸 = 𝑂)
4341, 29, 32, 42syl3anc 1475 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐸 = 𝑂)
4443oveq1d 6807 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → (𝐸 𝑉) = (𝑂 𝑉))
4522, 36, 443brtr4d 4816 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ ((𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)) ∧ ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊))) → 𝐼 (𝐸 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  Vcvv 3349   class class class wbr 4784  cfv 6031  crio 6752  (class class class)co 6792  Basecbs 16063  lecple 16155  joincjn 17151  meetcmee 17152  Atomscatm 35065  HLchlt 35152  LHypclh 35785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  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-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-undef 7550  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789
This theorem is referenced by:  cdleme26ee  36162
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