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Theorem cdleme28a 38007
Description: Lemma for cdleme25b 37991. 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 1285 . . . 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 37001 . . 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 1286 . . . 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 1205 . . . 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 1206 . . . 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 1208 . . . 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 1207 . . . 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 38003 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝐶𝐵)
213, 5, 6, 7, 8, 9, 20syl222anc 1387 . . 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 1209 . . . . 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 38003 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ 𝑃𝑄)) → 𝑌𝐵)
283, 5, 6, 7, 22, 9, 27syl222anc 1387 . . . 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 1204 . . . . . . 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 1129 . . . . . 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 1212 . . . . . 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 1210 . . . . . . 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 1299 . . . . . . 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 1300 . . . . . . 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 1129 . . . . . 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 37987 . . . . . 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 1372 . . . . 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 36926 . . . . 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 17777 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑉𝐵) → (𝑌 𝑉) ∈ 𝐵)
424, 28, 40, 41syl3anc 1372 . . 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 1301 . . . . 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 37635 . . . . . 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 17778 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
474, 43, 45, 46syl3anc 1372 . . . 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 17777 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑌 (𝑋 𝑊)) ∈ 𝐵)
494, 28, 47, 48syl3anc 1372 . . 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 37988 . . . . . 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 1372 . . . . 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 515 . . . 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 37986 . . . . . 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 1372 . . . . 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 515 . . . 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 38004 . . . 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 1402 . . 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 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))) → 𝑠𝐴)
59 simp23l 1295 . . . . . . 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 37004 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑠𝐴𝑡𝐴) → (𝑠 𝑡) ∈ 𝐵)
613, 58, 59, 60syl3anc 1372 . . . . . 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 17803 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑠 𝑡) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → ((𝑠 𝑡) (𝑋 𝑊)) (𝑋 𝑊))
634, 61, 47, 62syl3anc 1372 . . . . 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 5065 . . . 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 17792 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑉𝐵 ∧ (𝑋 𝑊) ∈ 𝐵𝑌𝐵)) → (𝑉 (𝑋 𝑊) → (𝑌 𝑉) (𝑌 (𝑋 𝑊))))
664, 40, 47, 28, 65syl13anc 1373 . . . 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 17784 . 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 17787 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑋 𝑊) (𝑌 (𝑋 𝑊)))
704, 28, 47, 69syl3anc 1372 . 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 17788 . . 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 1373 . 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 712 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wral 3053  ifcif 4414   class class class wbr 5030  cfv 6339  crio 7126  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  meetcmee 17671  Latclat 17771  Atomscatm 36900  HLchlt 36987  LHypclh 37621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-riotaBAD 36590
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-undef 7968  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135  df-lplanes 37136  df-lvols 37137  df-lines 37138  df-psubsp 37140  df-pmap 37141  df-padd 37433  df-lhyp 37625
This theorem is referenced by:  cdleme28b  38008
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