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Theorem cdleme26e 35964
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 1111 . . 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 1112 . . 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 1113 . . 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 1198 . . . 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 1200 . . . 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 553 . . 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 1116 . . 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 1228 . . . 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 1206 . . . 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 553 . . 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 1119 . . 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 35949 . . 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 1389 . 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 1199 . . . 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 1229 . . . 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 35962 . . . 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 1394 . . 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 1208 . . 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 1209 . . . 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 1207 . . . 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 553 . . 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 6239 . . . . 5 (Base‘𝐾) ∈ V
3425, 33eqeltri 2726 . . . 4 𝐵 ∈ V
3534, 26riotasv 34563 . . 3 ((𝐼𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐼 = 𝑁)
3628, 29, 32, 35syl3anc 1366 . 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 1201 . . . . 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 1230 . . . . 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 35962 . . . . 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 1394 . . . 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 34563 . . . 4 ((𝐸𝐵𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐸 = 𝑂)
4341, 29, 32, 42syl3anc 1366 . . 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 6705 . 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 4717 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 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:  cdleme26ee  35965
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