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Theorem cdleme26ee 34666
 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 1165 . . 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 1166 . . 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 1085 . . 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 1086 . . 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 1159 . . 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 34345 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑧𝐴𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)))
111, 2, 3, 4, 5, 10syl221anc 1329 . 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 1830 . . 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 2925 . . . . . 6 𝑧𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁)
15 nfcv 2751 . . . . . 6 𝑧𝐵
1614, 15nfriota 6520 . . . . 5 𝑧(𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
1713, 16nfcxfr 2749 . . . 4 𝑧𝐼
18 nfcv 2751 . . . 4 𝑧
19 cdleme26e.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
20 nfra1 2925 . . . . . . 7 𝑧𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂)
2120, 15nfriota 6520 . . . . . 6 𝑧(𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
2219, 21nfcxfr 2749 . . . . 5 𝑧𝐸
23 nfcv 2751 . . . . 5 𝑧
24 nfcv 2751 . . . . 5 𝑧𝑉
2522, 23, 24nfov 6575 . . . 4 𝑧(𝐸 𝑉)
2617, 18, 25nfbr 4629 . . 3 𝑧 𝐼 (𝐸 𝑉)
27 simp111 1183 . . . . 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 1184 . . . . 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 1185 . . . . 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 1186 . . . . 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 1187 . . . . 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 1188 . . . . 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 1169 . . . . 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 1170 . . . . . 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 1083 . . . . . 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 553 . . . . 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 1055 . . . . . 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 1082 . . . . . 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 553 . . . . 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 34665 . . . . 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 1350 . . . 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 1256 . . 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 3008 . 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LHypclh 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292 This theorem is referenced by:  cdleme27a  34673
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