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Theorem cdleme28a 37373
Description: Lemma for cdleme25b 37357. 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 1278 . . . 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)
43hllatd 36367 . . 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)
5 simp11r 1279 . . . 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)
6 simp12 1198 . . . 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))
7 simp13 1199 . . . 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))
8 simp22 1201 . . . 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))
9 simp21 1200 . . . 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)
10 cdleme26.j . . . . 5 = (join‘𝐾)
11 cdleme26.m . . . . 5 = (meet‘𝐾)
12 cdleme26.a . . . . 5 𝐴 = (Atoms‘𝐾)
13 cdleme26.h . . . . 5 𝐻 = (LHyp‘𝐾)
14 cdleme27.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
15 cdleme27.f . . . . 5 𝐹 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
16 cdleme27.z . . . . 5 𝑍 = ((𝑧 𝑈) (𝑄 ((𝑃 𝑧) 𝑊)))
17 cdleme27.n . . . . 5 𝑁 = ((𝑃 𝑄) (𝑍 ((𝑠 𝑧) 𝑊)))
18 cdleme27.d . . . . 5 𝐷 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑁))
19 cdleme27.c . . . . 5 𝐶 = if(𝑠 (𝑃 𝑄), 𝐷, 𝐹)
201, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme27cl 37369 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝐶𝐵)
213, 5, 6, 7, 8, 9, 20syl222anc 1380 . . 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)
22 simp23 1202 . . . . 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))
23 cdleme27.g . . . . . 6 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
24 cdleme27.o . . . . . 6 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
25 cdleme27.e . . . . . 6 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
26 cdleme27.y . . . . . 6 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
271, 2, 10, 11, 12, 13, 14, 23, 16, 24, 25, 26cdleme27cl 37369 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ 𝑃𝑄)) → 𝑌𝐵)
283, 5, 6, 7, 22, 9, 27syl222anc 1380 . . . 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)
29 simp11 1197 . . . . . . 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))
3029, 8, 223jca 1122 . . . . . 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)))
31 simp33 1205 . . . . . 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))
32 simp31 1203 . . . . . . 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)
33 simp32l 1292 . . . . . . 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)
34 simp32r 1293 . . . . . . 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)
3532, 33, 343jca 1122 . . . . . 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))
36 cdleme28a.v . . . . . . 7 𝑉 = ((𝑠 𝑡) (𝑋 𝑊))
371, 2, 10, 11, 12, 13, 36cdleme23b 37353 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑠𝑡 ∧ (𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → 𝑉𝐴)
3830, 31, 35, 37syl3anc 1365 . . . . 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)
391, 12atbase 36292 . . . . 5 (𝑉𝐴𝑉𝐵)
4038, 39syl 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)
411, 10latjcl 17651 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → (𝑌 𝑉) ∈ 𝐵)
424, 28, 40, 41syl3anc 1365 . . 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)
43 simp33l 1294 . . . . 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)
441, 13lhpbase 37001 . . . . . 6 (𝑊𝐻𝑊𝐵)
455, 44syl 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)
461, 11latmcl 17652 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
474, 43, 45, 46syl3anc 1365 . . . 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)
481, 10latjcl 17651 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑌 (𝑋 𝑊)) ∈ 𝐵)
494, 28, 47, 48syl3anc 1365 . . 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)
501, 2, 10, 11, 12, 13, 36cdleme23c 37354 . . . . . 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))
5130, 31, 35, 50syl3anc 1365 . . . . 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))
5232, 51jca 512 . . . 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)))
531, 2, 10, 11, 12, 13, 36cdleme23a 37352 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑠𝑡 ∧ (𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋)) → 𝑉 𝑊)
5430, 31, 35, 53syl3anc 1365 . . . . 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)
5538, 54jca 512 . . . 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))
561, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26cdleme27a 37370 . . . 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))
5729, 9, 8, 6, 7, 22, 52, 55, 56syl332anc 1395 . . 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))
58 simp22l 1286 . . . . . . 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)
59 simp23l 1288 . . . . . . 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)
601, 10, 12hlatjcl 36370 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑠𝐴𝑡𝐴) → (𝑠 𝑡) ∈ 𝐵)
613, 58, 59, 60syl3anc 1365 . . . . . 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)
621, 2, 11latmle2 17677 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑠 𝑡) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → ((𝑠 𝑡) (𝑋 𝑊)) (𝑋 𝑊))
634, 61, 47, 62syl3anc 1365 . . . . 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))
6436, 63eqbrtrid 5098 . . . 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))
651, 2, 10latjlej2 17666 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑉𝐵 ∧ (𝑋 𝑊) ∈ 𝐵𝑌𝐵)) → (𝑉 (𝑋 𝑊) → (𝑌 𝑉) (𝑌 (𝑋 𝑊))))
664, 40, 47, 28, 65syl13anc 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))) → (𝑉 (𝑋 𝑊) → (𝑌 𝑉) (𝑌 (𝑋 𝑊))))
6764, 66mpd 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)))
681, 2, 4, 21, 42, 49, 57, 67lattrd 17658 . 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)))
691, 2, 10latlej2 17661 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑋 𝑊) (𝑌 (𝑋 𝑊)))
704, 28, 47, 69syl3anc 1365 . 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)))
711, 2, 10latjle12 17662 . . 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))))
724, 21, 47, 49, 71syl13anc 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))) → ((𝐶 (𝑌 (𝑋 𝑊)) ∧ (𝑋 𝑊) (𝑌 (𝑋 𝑊))) ↔ (𝐶 (𝑋 𝑊)) (𝑌 (𝑋 𝑊))))
7368, 70, 72mpbi2and 708 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  ifcif 4470   class class class wbr 5063  cfv 6352  crio 7105  (class class class)co 7148  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  Latclat 17645  Atomscatm 36266  HLchlt 36353  LHypclh 36987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-riotaBAD 35956
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-undef 7930  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-p1 17640  df-lat 17646  df-clat 17708  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-llines 36501  df-lplanes 36502  df-lvols 36503  df-lines 36504  df-psubsp 36506  df-pmap 36507  df-padd 36799  df-lhyp 36991
This theorem is referenced by:  cdleme28b  37374
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