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Theorem cdleme28b 36525
Description: Lemma for cdleme25b 36508. TODO: FIX COMMENT. (Contributed by NM, 6-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
cdleme28b ((((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   𝑊,𝑠,𝑡,𝑢,𝑧   𝑋,𝑠,𝑧,𝑡
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)   𝑋(𝑢)   𝑌(𝑧,𝑢,𝑡,𝑠)   𝑍(𝑧,𝑢,𝑡,𝑠)

Proof of Theorem cdleme28b
StepHypRef Expression
1 cdleme26.b . 2 𝐵 = (Base‘𝐾)
2 cdleme26.l . 2 = (le‘𝐾)
3 simp11l 1340 . . 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 ∈ HL)
43hllatd 35518 . 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))) → 𝐾 ∈ Lat)
5 simp11r 1341 . . . 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 1218 . . . 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 1219 . . . 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 1221 . . . 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 1220 . . . 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 36520 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝐶𝐵)
213, 5, 6, 7, 8, 9, 20syl222anc 1454 . . 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 simp33l 1356 . . . 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))) → 𝑋𝐵)
231, 13lhpbase 36152 . . . . 5 (𝑊𝐻𝑊𝐵)
245, 23syl 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))) → 𝑊𝐵)
251, 11latmcl 17438 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
264, 22, 24, 25syl3anc 1439 . . 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))) → (𝑋 𝑊) ∈ 𝐵)
271, 10latjcl 17437 . . 3 ((𝐾 ∈ Lat ∧ 𝐶𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝐶 (𝑋 𝑊)) ∈ 𝐵)
284, 21, 26, 27syl3anc 1439 . 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 (𝑋 𝑊)) ∈ 𝐵)
29 simp23 1222 . . . 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))) → (𝑡𝐴 ∧ ¬ 𝑡 𝑊))
30 cdleme27.g . . . . 5 𝐺 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
31 cdleme27.o . . . . 5 𝑂 = ((𝑃 𝑄) (𝑍 ((𝑡 𝑧) 𝑊)))
32 cdleme27.e . . . . 5 𝐸 = (𝑢𝐵𝑧𝐴 ((¬ 𝑧 𝑊 ∧ ¬ 𝑧 (𝑃 𝑄)) → 𝑢 = 𝑂))
33 cdleme27.y . . . . 5 𝑌 = if(𝑡 (𝑃 𝑄), 𝐸, 𝐺)
341, 2, 10, 11, 12, 13, 14, 30, 16, 31, 32, 33cdleme27cl 36520 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ 𝑃𝑄)) → 𝑌𝐵)
353, 5, 6, 7, 29, 9, 34syl222anc 1454 . . 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))) → 𝑌𝐵)
361, 10latjcl 17437 . . 3 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑌 (𝑋 𝑊)) ∈ 𝐵)
374, 35, 26, 36syl3anc 1439 . 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))) → (𝑌 (𝑋 𝑊)) ∈ 𝐵)
38 eqid 2778 . . 3 ((𝑠 𝑡) (𝑋 𝑊)) = ((𝑠 𝑡) (𝑋 𝑊))
391, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 30, 31, 32, 33, 38cdleme28a 36524 . 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 (𝑋 𝑊)) (𝑌 (𝑋 𝑊)))
40 simp11 1217 . . 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 ∈ HL ∧ 𝑊𝐻))
41 simp31 1223 . . . 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))) → 𝑠𝑡)
4241necomd 3024 . . 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))) → 𝑡𝑠)
43 simp32 1224 . . . 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))) → ((𝑠 (𝑋 𝑊)) = 𝑋 ∧ (𝑡 (𝑋 𝑊)) = 𝑋))
4443ancomd 455 . . 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))) → ((𝑡 (𝑋 𝑊)) = 𝑋 ∧ (𝑠 (𝑋 𝑊)) = 𝑋))
45 simp33 1225 . . 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))) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
46 eqid 2778 . . . 4 ((𝑡 𝑠) (𝑋 𝑊)) = ((𝑡 𝑠) (𝑋 𝑊))
471, 2, 10, 11, 12, 13, 14, 30, 16, 31, 32, 33, 15, 17, 18, 19, 46cdleme28a 36524 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) ∧ (𝑡𝑠 ∧ ((𝑡 (𝑋 𝑊)) = 𝑋 ∧ (𝑠 (𝑋 𝑊)) = 𝑋) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝑌 (𝑋 𝑊)) (𝐶 (𝑋 𝑊)))
4840, 6, 7, 9, 29, 8, 42, 44, 45, 47syl333anc 1470 . 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))) → (𝑌 (𝑋 𝑊)) (𝐶 (𝑋 𝑊)))
491, 2, 4, 28, 37, 39, 48latasymd 17443 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  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wral 3090  ifcif 4307   class class class wbr 4886  cfv 6135  crio 6882  (class class class)co 6922  Basecbs 16255  lecple 16345  joincjn 17330  meetcmee 17331  Latclat 17431  Atomscatm 35417  HLchlt 35504  LHypclh 36138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-riotaBAD 35107
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-undef 7681  df-proset 17314  df-poset 17332  df-plt 17344  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-p0 17425  df-p1 17426  df-lat 17432  df-clat 17494  df-oposet 35330  df-ol 35332  df-oml 35333  df-covers 35420  df-ats 35421  df-atl 35452  df-cvlat 35476  df-hlat 35505  df-llines 35652  df-lplanes 35653  df-lvols 35654  df-lines 35655  df-psubsp 35657  df-pmap 35658  df-padd 35950  df-lhyp 36142
This theorem is referenced by:  cdleme28c  36526
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