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Theorem cdleme26ee 36168
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
cdleme26ee ((((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))) → 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   𝑧,𝑉
Allowed substitution hints:   𝐸(𝑧,𝑢)   𝐹(𝑧,𝑢)   𝐻(𝑢)   𝐼(𝑧,𝑢)   𝐾(𝑢)   𝑁(𝑧)   𝑂(𝑧)   𝑉(𝑢)

Proof of Theorem cdleme26ee
StepHypRef Expression
1 simp11l 1368 . . 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))) → 𝐾 ∈ HL)
2 simp11r 1369 . . 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))) → 𝑊𝐻)
3 simp12 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))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp13 1247 . . 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))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5 simp3l1 1362 . . 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))) → 𝑃𝑄)
6 cdleme26.l . . . 4 = (le‘𝐾)
7 cdleme26.j . . . 4 = (join‘𝐾)
8 cdleme26.a . . . 4 𝐴 = (Atoms‘𝐾)
9 cdleme26.h . . . 4 𝐻 = (LHyp‘𝐾)
106, 7, 8, 9cdlemb2 35848 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑧𝐴𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))
111, 2, 3, 4, 5, 10syl221anc 1487 . 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))) → ∃𝑧𝐴𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))
12 nfv 1995 . . 3 𝑧(((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)))
13 cdleme26e.i . . . . 5 𝐼 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
14 nfra1 3090 . . . . . 6 𝑧𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁)
15 nfcv 2913 . . . . . 6 𝑧𝐵
1614, 15nfriota 6766 . . . . 5 𝑧(𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
1713, 16nfcxfr 2911 . . . 4 𝑧𝐼
18 nfcv 2913 . . . 4 𝑧
19 cdleme26e.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
20 nfra1 3090 . . . . . . 7 𝑧𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)
2120, 15nfriota 6766 . . . . . 6 𝑧(𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
2219, 21nfcxfr 2911 . . . . 5 𝑧𝐸
23 nfcv 2913 . . . . 5 𝑧
24 nfcv 2913 . . . . 5 𝑧𝑉
2522, 23, 24nfov 6825 . . . 4 𝑧(𝐸 𝑉)
2617, 18, 25nfbr 4834 . . 3 𝑧 𝐼 (𝐸 𝑉)
27 simp111 1386 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
28 simp112 1387 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
29 simp113 1388 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
30 simp121 1389 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
31 simp122 1390 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
32 simp123 1391 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑉𝐴𝑉 𝑊))
33 simp13l 1372 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑃𝑄𝑆 (𝑃 𝑄) ∧ 𝑇 (𝑃 𝑄)))
34 simp13r 1373 . . . . . 6 (((((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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑇 𝑉) = (𝑃 𝑄))
35 simp3r 1244 . . . . . 6 (((((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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → ¬ 𝑧 (𝑃 𝑄))
3634, 35jca 501 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → ((𝑇 𝑉) = (𝑃 𝑄) ∧ ¬ 𝑧 (𝑃 𝑄)))
37 simp2 1131 . . . . . 6 (((((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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝑧𝐴)
38 simp3l 1243 . . . . . 6 (((((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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → ¬ 𝑧 𝑊)
3937, 38jca 501 . . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → (𝑧𝐴 ∧ ¬ 𝑧 𝑊))
40 cdleme26.b . . . . . 6 𝐵 = (Base‘𝐾)
41 cdleme26.m . . . . . 6 = (meet‘𝐾)
42 cdleme26e.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
43 cdleme26e.f . . . . . 6 𝐹 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
44 cdleme26e.n . . . . . 6 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑧) 𝑊)))
45 cdleme26e.o . . . . . 6 𝑂 = ((𝑃 𝑄) (𝐹 ((𝑇 𝑧) 𝑊)))
4640, 6, 7, 41, 8, 9, 42, 43, 44, 45, 13, 19cdleme26e 36167 . . . . 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))) → 𝐼 (𝐸 𝑉))
4727, 28, 29, 30, 31, 32, 33, 36, 39, 46syl333anc 1508 . . . 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))) ∧ 𝑧𝐴 ∧ (¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄))) → 𝐼 (𝐸 𝑉))
48473exp 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))) → (𝑧𝐴 → ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝐼 (𝐸 𝑉))))
4912, 26, 48rexlimd 3174 . 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))) → (∃𝑧𝐴𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝐼 (𝐸 𝑉)))
5011, 49mpd 15 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))) → 𝐼 (𝐸 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062   class class class wbr 4787  cfv 6030  crio 6756  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35070  HLchlt 35157  LHypclh 35791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-riotaBAD 34759
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-undef 7555  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34983  df-ol 34985  df-oml 34986  df-covers 35073  df-ats 35074  df-atl 35105  df-cvlat 35129  df-hlat 35158  df-llines 35305  df-lplanes 35306  df-lvols 35307  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795
This theorem is referenced by:  cdleme27a  36175
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