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Theorem cdleme28a 35975
Description: Lemma for cdleme25b 35959. TODO: FIX COMMENT. (Contributed by NM, 4-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)
cdleme28a.v 𝑉 = ((𝑠 𝑡) (𝑋 𝑊))
Assertion
Ref Expression
cdleme28a ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐶 (𝑋 𝑊)) (𝑌 (𝑋 𝑊)))
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   𝑋,𝑠
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)   𝑋(𝑧,𝑢,𝑡)   𝑌(𝑧,𝑢,𝑡,𝑠)   𝑍(𝑧,𝑢,𝑡,𝑠)

Proof of Theorem cdleme28a
StepHypRef Expression
1 cdleme26.b . . 3 𝐵 = (Base‘𝐾)
2 cdleme26.l . . 3 = (le‘𝐾)
3 simp11l 1192 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝐾 ∈ HL)
4 hllat 34968 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝐾 ∈ Lat)
6 simp11r 1193 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑊𝐻)
7 simp12 1112 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
8 simp13 1113 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
9 simp22 1115 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
10 simp21 1114 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑃𝑄)
11 cdleme26.j . . . . 5 = (join‘𝐾)
12 cdleme26.m . . . . 5 = (meet‘𝐾)
13 cdleme26.a . . . . 5 𝐴 = (Atoms‘𝐾)
14 cdleme26.h . . . . 5 𝐻 = (LHyp‘𝐾)
15 cdleme27.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
16 cdleme27.f . . . . 5 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
17 cdleme27.z . . . . 5 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
18 cdleme27.n . . . . 5 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
19 cdleme27.d . . . . 5 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
20 cdleme27.c . . . . 5 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
211, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20cdleme27cl 35971 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝐶𝐵)
223, 6, 7, 8, 9, 10, 21syl222anc 1382 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝐶𝐵)
23 simp23 1116 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
24 cdleme27.g . . . . . 6 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
25 cdleme27.o . . . . . 6 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
26 cdleme27.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
27 cdleme27.y . . . . . 6 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
281, 2, 11, 12, 13, 14, 15, 24, 17, 25, 26, 27cdleme27cl 35971 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ 𝑃𝑄)) → 𝑌𝐵)
293, 6, 7, 8, 23, 10, 28syl222anc 1382 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑌𝐵)
30 simp11 1111 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3130, 9, 233jca 1261 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)))
32 simp33 1119 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
33 simp31 1117 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑠𝑡)
34 simp32l 1206 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑠 (𝑋 𝑊)) = 𝑋)
35 simp32r 1207 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑡 (𝑋 𝑊)) = 𝑋)
3633, 34, 353jca 1261 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑠𝑡 ∧ (𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋))
37 cdleme28a.v . . . . . . 7 𝑉 = ((𝑠 𝑡) (𝑋 𝑊))
381, 2, 11, 12, 13, 14, 37cdleme23b 35955 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑠𝑡 ∧ (𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → 𝑉𝐴)
3931, 32, 36, 38syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑉𝐴)
401, 13atbase 34894 . . . . 5 (𝑉𝐴𝑉𝐵)
4139, 40syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑉𝐵)
421, 11latjcl 17098 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → (𝑌 𝑉) ∈ 𝐵)
435, 29, 41, 42syl3anc 1366 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑌 𝑉) ∈ 𝐵)
44 simp33l 1208 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑋𝐵)
451, 14lhpbase 35602 . . . . . 6 (𝑊𝐻𝑊𝐵)
466, 45syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑊𝐵)
471, 12latmcl 17099 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
485, 44, 46, 47syl3anc 1366 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑋 𝑊) ∈ 𝐵)
491, 11latjcl 17098 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑌 (𝑋 𝑊)) ∈ 𝐵)
505, 29, 48, 49syl3anc 1366 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑌 (𝑋 𝑊)) ∈ 𝐵)
511, 2, 11, 12, 13, 14, 37cdleme23c 35956 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑠𝑡 ∧ (𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → 𝑠 (𝑡 𝑉))
5231, 32, 36, 51syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑠 (𝑡 𝑉))
5333, 52jca 553 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑠𝑡𝑠 (𝑡 𝑉)))
541, 2, 11, 12, 13, 14, 37cdleme23a 35954 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑠𝑡 ∧ (𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → 𝑉 𝑊)
5531, 32, 36, 54syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑉 𝑊)
5639, 55jca 553 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑉𝐴𝑉 𝑊))
571, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27cdleme27a 35972 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ ((𝑠𝑡𝑠 (𝑡 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐶 (𝑌 𝑉))
5830, 10, 9, 7, 8, 23, 53, 56, 57syl332anc 1397 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝐶 (𝑌 𝑉))
59 simp22l 1200 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑠𝐴)
60 simp23l 1202 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑡𝐴)
611, 11, 13hlatjcl 34971 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑠𝐴𝑡𝐴) → (𝑠 𝑡) ∈ 𝐵)
623, 59, 60, 61syl3anc 1366 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑠 𝑡) ∈ 𝐵)
631, 2, 12latmle2 17124 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑠 𝑡) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → ((𝑠 𝑡) (𝑋 𝑊)) (𝑋 𝑊))
645, 62, 48, 63syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → ((𝑠 𝑡) (𝑋 𝑊)) (𝑋 𝑊))
6537, 64syl5eqbr 4720 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝑉 (𝑋 𝑊))
661, 2, 11latjlej2 17113 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑉𝐵 ∧ (𝑋 𝑊) ∈ 𝐵𝑌𝐵)) → (𝑉 (𝑋 𝑊) → (𝑌 𝑉) (𝑌 (𝑋 𝑊))))
675, 41, 48, 29, 66syl13anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑉 (𝑋 𝑊) → (𝑌 𝑉) (𝑌 (𝑋 𝑊))))
6865, 67mpd 15 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑌 𝑉) (𝑌 (𝑋 𝑊)))
691, 2, 5, 22, 43, 50, 58, 68lattrd 17105 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → 𝐶 (𝑌 (𝑋 𝑊)))
701, 2, 11latlej2 17108 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑋 𝑊) (𝑌 (𝑋 𝑊)))
715, 29, 48, 70syl3anc 1366 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑋 𝑊) (𝑌 (𝑋 𝑊)))
721, 2, 11latjle12 17109 . . 3 ((𝐾 ∈ Lat ∧ (𝐶𝐵 ∧ (𝑋 𝑊) ∈ 𝐵 ∧ (𝑌 (𝑋 𝑊)) ∈ 𝐵)) → ((𝐶 (𝑌 (𝑋 𝑊)) ∧ (𝑋 𝑊) (𝑌 (𝑋 𝑊))) ↔ (𝐶 (𝑋 𝑊)) (𝑌 (𝑋 𝑊))))
735, 22, 48, 50, 72syl13anc 1368 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → ((𝐶 (𝑌 (𝑋 𝑊)) ∧ (𝑋 𝑊) (𝑌 (𝑋 𝑊))) ↔ (𝐶 (𝑋 𝑊)) (𝑌 (𝑋 𝑊))))
7469, 71, 73mpbi2and 976 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑠𝑡 ∧ ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐶 (𝑋 𝑊)) (𝑌 (𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  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  Latclat 17092  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:  cdleme28b  35976
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