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Theorem cdleme27a 36176
Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 36172 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 1396 . . . . . . 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 1399 . . . . . . 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 1400 . . . . . . 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 1398 . . . . . . 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 1401 . . . . . . 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 1380 . . . . . . 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 1397 . . . . . . . 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 1240 . . . . . . . 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 1241 . . . . . . . 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 1123 . . . . . . 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 1133 . . . . . . 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 36169 . . . . . . 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 1508 . . . . . 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 1115 . . . . 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 1241 . . . . . . 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 1369 . . . . . . . . 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 1129 . . . . . . . 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 1373 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝐴)
31303ad2ant2 1129 . . . . . . . . 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 1379 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑡𝐴)
33323ad2ant2 1129 . . . . . . . . 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 1311 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝑡)
35343ad2ant2 1129 . . . . . . . . 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 1123 . . . . . . . 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 1375 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝐴)
38373ad2ant2 1129 . . . . . . . 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 1377 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑄𝐴)
40393ad2ant2 1129 . . . . . . . 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 1397 . . . . . . . 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 1313 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐴)
43423ad2ant2 1129 . . . . . . . 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 1133 . . . . . . . . 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 1312 . . . . . . . . . 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 1129 . . . . . . . . 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 1240 . . . . . . . . 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 1123 . . . . . . . 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 36150 . . . . . . . 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 1504 . . . . . . 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 186 . . . . . 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 1115 . . . . 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 3019 . . . 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 4237 . . . . . 6 (𝑠 (𝑃 𝑄) → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = 𝐷)
5654, 55syl5eq 2807 . . . . 5 (𝑠 (𝑃 𝑄) → 𝐶 = 𝐷)
5756ad2antrr 764 . . . 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 4237 . . . . . . 7 (𝑡 (𝑃 𝑄) → if(𝑡 (𝑃 𝑄), 𝐸, 𝐺) = 𝐸)
6058, 59syl5eq 2807 . . . . . 6 (𝑡 (𝑃 𝑄) → 𝑌 = 𝐸)
6160oveq1d 6830 . . . . 5 (𝑡 (𝑃 𝑄) → (𝑌 𝑉) = (𝐸 𝑉))
6261ad2antlr 765 . . . 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 4837 . . 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 449 . 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 1333 . . . . 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 1335 . . . . . 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 807 . . . . . 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 555 . . . . 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 1343 . . . . 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 1339 . . . . 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 1341 . . . . 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 1337 . . . . 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 809 . . . . 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 1299 . . . . 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 1301 . . . . 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 2761 . . . . . 6 ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊)))
78 eqid 2761 . . . . . . 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 36167 . . . . . 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 36172 . . . . 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 1509 . . . 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 764 . . . 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 4240 . . . . . . 7 𝑡 (𝑃 𝑄) → if(𝑡 (𝑃 𝑄), 𝐸, 𝐺) = 𝐺)
8458, 83syl5eq 2807 . . . . . 6 𝑡 (𝑃 𝑄) → 𝑌 = 𝐺)
8584oveq1d 6830 . . . . 5 𝑡 (𝑃 𝑄) → (𝑌 𝑉) = (𝐺 𝑉))
8685ad2antlr 765 . . . 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 4837 . . 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 449 . 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 1333 . . . . 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 1335 . . . . . 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 809 . . . . . 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 555 . . . . 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 1337 . . . . 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 1339 . . . . 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 1341 . . . . 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 1343 . . . . 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 807 . . . . 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 1299 . . . . 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 1301 . . . . 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 2761 . . . . . 6 ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊)))
102 eqid 2761 . . . . . . 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 36167 . . . . . 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 36174 . . . . 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 1509 . . . 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 4240 . . . . . 6 𝑠 (𝑃 𝑄) → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = 𝐹)
10754, 106syl5eq 2807 . . . . 5 𝑠 (𝑃 𝑄) → 𝐶 = 𝐹)
108107ad2antrr 764 . . . 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 765 . . . 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 4837 . . 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 449 . 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 1333 . . . . 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 1343 . . . . 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 809 . . . . . 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 807 . . . . . 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 1335 . . . . . 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 1123 . . . . 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 1339 . . . . 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 1341 . . . . 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 1337 . . . . 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 1299 . . . . 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 1301 . . . . 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 36157 . . . . 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 1507 . . . 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 764 . . . 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 765 . . . 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 4837 . . 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 449 . 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 1027 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 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933  wral 3051  ifcif 4231   class class class wbr 4805  cfv 6050  crio 6775  (class class class)co 6815  Basecbs 16080  lecple 16171  joincjn 17166  meetcmee 17167  Atomscatm 35072  HLchlt 35159  LHypclh 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-riotaBAD 34761
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-undef 7570  df-preset 17150  df-poset 17168  df-plt 17180  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-p0 17261  df-p1 17262  df-lat 17268  df-clat 17330  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308  df-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796
This theorem is referenced by:  cdleme27N  36178  cdleme28a  36179
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