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Theorem cdleme27a 35972
Description: Part of proof of Lemma E in [Crawley] p. 113. cdleme26f 35968 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 1219 . . . . . . 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 1222 . . . . . . 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 1223 . . . . . . 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 1221 . . . . . . 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 1224 . . . . . . 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 1203 . . . . . . 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 1220 . . . . . . . 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 1105 . . . . . . . 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 1106 . . . . . . . 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 1261 . . . . . . 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 1083 . . . . . . 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 35965 . . . . . . 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 1397 . . . . . 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 1286 . . . . 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 1106 . . . . . . 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 1192 . . . . . . . . 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 1103 . . . . . . . 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 1196 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝐴)
31303ad2ant2 1103 . . . . . . . . 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 1202 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑡𝐴)
33323ad2ant2 1103 . . . . . . . . 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 1152 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑠𝑡)
35343ad2ant2 1103 . . . . . . . . 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 1261 . . . . . . . 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 1198 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝐴)
38373ad2ant2 1103 . . . . . . . 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 1200 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑄𝐴)
40393ad2ant2 1103 . . . . . . . 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 1220 . . . . . . . 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 1154 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐴)
43423ad2ant2 1103 . . . . . . . 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 1083 . . . . . . . . 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 1153 . . . . . . . . . 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 1103 . . . . . . . . 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 1105 . . . . . . . . 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 1261 . . . . . . . 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 35946 . . . . . . . 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 1393 . . . . . . 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 1286 . . . . 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 2909 . . . 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 4125 . . . . . 6 (𝑠 (𝑃 𝑄) → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = 𝐷)
5654, 55syl5eq 2697 . . . . 5 (𝑠 (𝑃 𝑄) → 𝐶 = 𝐷)
5756ad2antrr 762 . . . 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 4125 . . . . . . 7 (𝑡 (𝑃 𝑄) → if(𝑡 (𝑃 𝑄), 𝐸, 𝐺) = 𝐸)
6058, 59syl5eq 2697 . . . . . 6 (𝑡 (𝑃 𝑄) → 𝑌 = 𝐸)
6160oveq1d 6705 . . . . 5 (𝑡 (𝑃 𝑄) → (𝑌 𝑉) = (𝐸 𝑉))
6261ad2antlr 763 . . . 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 4717 . . 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 1165 . . . . 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 1166 . . . . . 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 805 . . . . . 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 553 . . . . 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 1170 . . . . 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 1168 . . . . 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 1169 . . . . 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 1167 . . . . 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 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)))) → ¬ 𝑡 (𝑃 𝑄))
74 simpr3l 1142 . . . . 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 1143 . . . . 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 2651 . . . . . 6 ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐺 ((𝑠 𝑡) 𝑊)))
78 eqid 2651 . . . . . . 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 35963 . . . . . 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 35968 . . . . 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 1398 . . . 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 762 . . . 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 4128 . . . . . . 7 𝑡 (𝑃 𝑄) → if(𝑡 (𝑃 𝑄), 𝐸, 𝐺) = 𝐺)
8458, 83syl5eq 2697 . . . . . 6 𝑡 (𝑃 𝑄) → 𝑌 = 𝐺)
8584oveq1d 6705 . . . . 5 𝑡 (𝑃 𝑄) → (𝑌 𝑉) = (𝐺 𝑉))
8685ad2antlr 763 . . . 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 4717 . . 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 1165 . . . . 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 1166 . . . . . 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 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)))) → 𝑡 (𝑃 𝑄))
9290, 91jca 553 . . . . 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 1167 . . . . 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 1168 . . . . 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 1169 . . . . 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 1170 . . . . 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 805 . . . . 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 1142 . . . . 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 1143 . . . . 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 2651 . . . . . 6 ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊))) = ((𝑃 𝑄) (𝐹 ((𝑡 𝑠) 𝑊)))
102 eqid 2651 . . . . . . 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 35963 . . . . . 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 35970 . . . . 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 1398 . . . 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 4128 . . . . . 6 𝑠 (𝑃 𝑄) → if(𝑠 (𝑃 𝑄), 𝐷, 𝐹) = 𝐹)
10754, 106syl5eq 2697 . . . . 5 𝑠 (𝑃 𝑄) → 𝐶 = 𝐹)
108107ad2antrr 762 . . . 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 763 . . . 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 4717 . . 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 1165 . . . . 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 1170 . . . . 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 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)))) → ¬ 𝑡 (𝑃 𝑄))
115 simpll 805 . . . . . 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 1166 . . . . . 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 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)))) → (¬ 𝑡 (𝑃 𝑄) ∧ ¬ 𝑠 (𝑃 𝑄) ∧ 𝑃𝑄))
118 simpr21 1168 . . . . 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 1169 . . . . 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 1167 . . . . 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 1142 . . . . 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 1143 . . . . 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 35953 . . . . 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 1396 . . . 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 762 . . . 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 763 . . . 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 4717 . . 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 1009 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 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  ifcif 4119   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:  cdleme27N  35974  cdleme28a  35975
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