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Theorem cdleme27a 38830
Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 38826 with s and t swapped (this case is not mentioned by them). If s t v, then f(s) fs(t) v. TODO: FIX COMMENT. (Contributed by NM, 3-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)
cdleme27.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme27.f 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme27.z 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
cdleme27.n 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
cdleme27.d 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
cdleme27.c 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
cdleme27.g 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme27.o 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
cdleme27.e 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
cdleme27.y 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
Assertion
Ref Expression
cdleme27a ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
Distinct variable groups:   𝑡,𝑠,𝑢,𝑧,𝐴   𝐵,𝑠,𝑡,𝑢,𝑧   𝑢,𝐹   𝑢,𝐺   𝐻,𝑠,𝑡,𝑧   ,𝑠,𝑡,𝑢,𝑧   𝐾,𝑠,𝑡,𝑧   ,𝑠,𝑡,𝑢,𝑧   ,𝑠,𝑡,𝑢,𝑧   𝑡,𝑁,𝑢   𝑂,𝑠,𝑢   𝑃,𝑠,𝑡,𝑢,𝑧   𝑄,𝑠,𝑡,𝑢,𝑧   𝑈,𝑠,𝑡,𝑢,𝑧   𝑧,𝑉   𝑊,𝑠,𝑡,𝑢,𝑧
Allowed substitution hints:   𝐶(𝑧,𝑢,𝑡,𝑠)   𝐷(𝑧,𝑢,𝑡,𝑠)   𝐸(𝑧,𝑢,𝑡,𝑠)   𝐹(𝑧,𝑡,𝑠)   𝐺(𝑧,𝑡,𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑁(𝑧,𝑠)   𝑂(𝑧,𝑡)   𝑉(𝑢,𝑡,𝑠)   𝑌(𝑧,𝑢,𝑡,𝑠)   𝑍(𝑧,𝑢,𝑡,𝑠)

Proof of Theorem cdleme27a
StepHypRef Expression
1 simp211 1311 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp221 1314 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp222 1315 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp213 1313 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
5 simp223 1316 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
6 simp23r 1295 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑉𝐴𝑉 𝑊))
7 simp212 1312 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → 𝑃𝑄)
8 simp1l 1197 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → 𝑠 (𝑃 𝑄))
9 simp1r 1198 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → 𝑡 (𝑃 𝑄))
107, 8, 93jca 1128 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑃𝑄𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)))
11 simp3 1138 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → (𝑡 𝑉) = (𝑃 𝑄))
12 cdleme26.b . . . . . . . 8 𝐵 = (Base‘𝐾)
13 cdleme26.l . . . . . . . 8 = (le‘𝐾)
14 cdleme26.j . . . . . . . 8 = (join‘𝐾)
15 cdleme26.m . . . . . . . 8 = (meet‘𝐾)
16 cdleme26.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
17 cdleme26.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
18 cdleme27.u . . . . . . . 8 𝑈 = ((𝑃 𝑄) 𝑊)
19 cdleme27.z . . . . . . . 8 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
20 cdleme27.n . . . . . . . 8 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
21 cdleme27.o . . . . . . . 8 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
22 cdleme27.d . . . . . . . 8 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
23 cdleme27.e . . . . . . . 8 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
2412, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23cdleme26ee 38823 . . . . . . 7 ((((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))) → 𝐷 (𝐸 𝑉))
251, 2, 3, 4, 5, 6, 10, 11, 24syl332anc 1401 . . . . . 6 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) = (𝑃 𝑄)) → 𝐷 (𝐸 𝑉))
26253expia 1121 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → ((𝑡 𝑉) = (𝑃 𝑄) → 𝐷 (𝐸 𝑉)))
27 simp1r 1198 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑡 (𝑃 𝑄))
28 simp11l 1284 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐾 ∈ HL)
29283ad2ant2 1134 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝐾 ∈ HL)
30 simp13l 1288 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝐴)
31303ad2ant2 1134 . . . . . . . . 9 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑠𝐴)
32 simp23l 1294 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑡𝐴)
33323ad2ant2 1134 . . . . . . . . 9 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑡𝐴)
34 simp3ll 1244 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝑡)
35343ad2ant2 1134 . . . . . . . . 9 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑠𝑡)
3631, 33, 353jca 1128 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → (𝑠𝐴𝑡𝐴𝑠𝑡))
37 simp21l 1290 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝐴)
38373ad2ant2 1134 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑃𝐴)
39 simp22l 1292 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑄𝐴)
40393ad2ant2 1134 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑄𝐴)
41 simp212 1312 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑃𝑄)
42 simp3rl 1246 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐴)
43423ad2ant2 1134 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑉𝐴)
44 simp3 1138 . . . . . . . . 9 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → (𝑡 𝑉) ≠ (𝑃 𝑄))
45 simp3lr 1245 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠 (𝑡 𝑉))
46453ad2ant2 1134 . . . . . . . . 9 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑠 (𝑡 𝑉))
47 simp1l 1197 . . . . . . . . 9 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝑠 (𝑃 𝑄))
4844, 46, 473jca 1128 . . . . . . . 8 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → ((𝑡 𝑉) ≠ (𝑃 𝑄) ∧ 𝑠 (𝑡 𝑉) ∧ 𝑠 (𝑃 𝑄)))
4913, 14, 15, 16, 17cdleme22b 38804 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑠𝐴𝑡𝐴𝑠𝑡)) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑉𝐴 ∧ ((𝑡 𝑉) ≠ (𝑃 𝑄) ∧ 𝑠 (𝑡 𝑉) ∧ 𝑠 (𝑃 𝑄)))) → ¬ 𝑡 (𝑃 𝑄))
5029, 36, 38, 40, 41, 43, 48, 49syl232anc 1397 . . . . . . 7 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → ¬ 𝑡 (𝑃 𝑄))
5127, 50pm2.21dd 194 . . . . . 6 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) ∧ (𝑡 𝑉) ≠ (𝑃 𝑄)) → 𝐷 (𝐸 𝑉))
52513expia 1121 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → ((𝑡 𝑉) ≠ (𝑃 𝑄) → 𝐷 (𝐸 𝑉)))
5326, 52pm2.61dne 3031 . . . 4 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐷 (𝐸 𝑉))
54 cdleme27.c . . . . . 6 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
55 iftrue 4492 . . . . . 6 (𝑠 (𝑃 𝑄) → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = 𝐷)
5654, 55eqtrid 2788 . . . . 5 (𝑠 (𝑃 𝑄) → 𝐶 = 𝐷)
5756ad2antrr 724 . . . 4 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 = 𝐷)
58 cdleme27.y . . . . . . 7 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
59 iftrue 4492 . . . . . . 7 (𝑡 (𝑃 𝑄) → if(𝑡 (𝑃 𝑄), 𝐸, 𝐺) = 𝐸)
6058, 59eqtrid 2788 . . . . . 6 (𝑡 (𝑃 𝑄) → 𝑌 = 𝐸)
6160oveq1d 7372 . . . . 5 (𝑡 (𝑃 𝑄) → (𝑌 𝑉) = (𝐸 𝑉))
6261ad2antlr 725 . . . 4 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑌 𝑉) = (𝐸 𝑉))
6353, 57, 623brtr4d 5137 . . 3 (((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 (𝑌 𝑉))
6463ex 413 . 2 ((𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉)))
65 simpr11 1257 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
66 simpr12 1258 . . . . . 6 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝑃𝑄)
67 simpll 765 . . . . . 6 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝑠 (𝑃 𝑄))
6866, 67jca 512 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑃𝑄𝑠 (𝑃 𝑄)))
69 simpr23 1262 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
70 simpr21 1260 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
71 simpr22 1261 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
72 simpr13 1259 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
73 simplr 767 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → ¬ 𝑡 (𝑃 𝑄))
74 simpr3l 1234 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑠𝑡𝑠 (𝑡 𝑉)))
75 simpr3r 1235 . . . . 5 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑉𝐴𝑉 𝑊))
76 cdleme27.g . . . . . 6 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
77 eqid 2736 . . . . . 6 ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊)))
78 eqid 2736 . . . . . . 7 (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊))))) = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊)))))
7919, 20, 76, 77, 22, 78cdleme25cv 38821 . . . . . 6 𝐷 = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊)))))
8012, 13, 14, 15, 16, 17, 18, 76, 77, 79cdleme26f 38826 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑠 (𝑃 𝑄)) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ (𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐷 (𝐺 𝑉))
8165, 68, 69, 70, 71, 72, 73, 74, 75, 80syl333anc 1402 . . . 4 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐷 (𝐺 𝑉))
8256ad2antrr 724 . . . 4 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 = 𝐷)
83 iffalse 4495 . . . . . . 7 𝑡 (𝑃 𝑄) → if(𝑡 (𝑃 𝑄), 𝐸, 𝐺) = 𝐺)
8458, 83eqtrid 2788 . . . . . 6 𝑡 (𝑃 𝑄) → 𝑌 = 𝐺)
8584oveq1d 7372 . . . . 5 𝑡 (𝑃 𝑄) → (𝑌 𝑉) = (𝐺 𝑉))
8685ad2antlr 725 . . . 4 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑌 𝑉) = (𝐺 𝑉))
8781, 82, 863brtr4d 5137 . . 3 (((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 (𝑌 𝑉))
8887ex 413 . 2 ((𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉)))
89 simpr11 1257 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
90 simpr12 1258 . . . . . 6 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝑃𝑄)
91 simplr 767 . . . . . 6 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝑡 (𝑃 𝑄))
9290, 91jca 512 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑃𝑄𝑡 (𝑃 𝑄)))
93 simpr13 1259 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
94 simpr21 1260 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
95 simpr22 1261 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
96 simpr23 1262 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
97 simpll 765 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → ¬ 𝑠 (𝑃 𝑄))
98 simpr3l 1234 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑠𝑡𝑠 (𝑡 𝑉)))
99 simpr3r 1235 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑉𝐴𝑉 𝑊))
100 cdleme27.f . . . . . 6 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
101 eqid 2736 . . . . . 6 ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊)))
102 eqid 2736 . . . . . . 7 (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊))))) = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊)))))
10319, 21, 100, 101, 23, 102cdleme25cv 38821 . . . . . 6 𝐸 = (𝑢𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) → 𝑢 = ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊)))))
10412, 13, 14, 15, 16, 17, 18, 100, 101, 103cdleme26f2 38828 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝑄𝑡 (𝑃 𝑄)) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (¬ 𝑠 (𝑃 𝑄) ∧ (𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐹 (𝐸 𝑉))
10589, 92, 93, 94, 95, 96, 97, 98, 99, 104syl333anc 1402 . . . 4 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐹 (𝐸 𝑉))
106 iffalse 4495 . . . . . 6 𝑠 (𝑃 𝑄) → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = 𝐹)
10754, 106eqtrid 2788 . . . . 5 𝑠 (𝑃 𝑄) → 𝐶 = 𝐹)
108107ad2antrr 724 . . . 4 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 = 𝐹)
10961ad2antlr 725 . . . 4 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑌 𝑉) = (𝐸 𝑉))
110105, 108, 1093brtr4d 5137 . . 3 (((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 (𝑌 𝑉))
111110ex 413 . 2 ((¬ 𝑠 (𝑃 𝑄) ∧ 𝑡 (𝑃 𝑄)) → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉)))
112 simpr11 1257 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
113 simpr23 1262 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
114 simplr 767 . . . . . 6 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → ¬ 𝑡 (𝑃 𝑄))
115 simpll 765 . . . . . 6 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → ¬ 𝑠 (𝑃 𝑄))
116 simpr12 1258 . . . . . 6 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝑃𝑄)
117114, 115, 1163jca 1128 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (¬ 𝑡 (𝑃 𝑄) ∧ ¬ 𝑠 (𝑃 𝑄) ∧ 𝑃𝑄))
118 simpr21 1260 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
119 simpr22 1261 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
120 simpr13 1259 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
121 simpr3l 1234 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑠𝑡𝑠 (𝑡 𝑉)))
122 simpr3r 1235 . . . . 5 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑉𝐴𝑉 𝑊))
12313, 14, 15, 16, 17, 18, 100, 76cdleme22g 38811 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ (¬ 𝑡 (𝑃 𝑄) ∧ ¬ 𝑠 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐹 (𝐺 𝑉))
124112, 113, 117, 118, 119, 120, 121, 122, 123syl323anc 1400 . . . 4 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐹 (𝐺 𝑉))
125107ad2antrr 724 . . . 4 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 = 𝐹)
12685ad2antlr 725 . . . 4 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → (𝑌 𝑉) = (𝐺 𝑉))
127124, 125, 1263brtr4d 5137 . . 3 (((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊)))) → 𝐶 (𝑌 𝑉))
128127ex 413 . 2 ((¬ 𝑠 (𝑃 𝑄) ∧ ¬ 𝑡 (𝑃 𝑄)) → ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉)))
12964, 88, 111, 1284cases 1039 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  ifcif 4486   class class class wbr 5105  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Atomscatm 37725  HLchlt 37812  LHypclh 38447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-undef 8204  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451
This theorem is referenced by:  cdleme27N  38832  cdleme28a  38833
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