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Theorem cdleme32e 39855
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
cdleme32.b 𝐵 = (Base‘𝐾)
cdleme32.l = (le‘𝐾)
cdleme32.j = (join‘𝐾)
cdleme32.m = (meet‘𝐾)
cdleme32.a 𝐴 = (Atoms‘𝐾)
cdleme32.h 𝐻 = (LHyp‘𝐾)
cdleme32.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme32.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme32.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme32.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdleme32.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
cdleme32.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
cdleme32.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme32.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
Assertion
Ref Expression
cdleme32e ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝑦,𝐶   𝐷,𝑠,𝑦,𝑧   𝑦,𝐸   𝐻,𝑠,𝑡   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑥,𝑁,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑋,𝑠,𝑡,𝑥,𝑧   𝑦,𝐻   𝑦,𝐾   𝑦,𝑌   𝑧,𝐻   𝑧,𝐾   𝑌,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑧,𝑡,𝑠)   𝐷(𝑥,𝑡)   𝐸(𝑥,𝑧,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑋(𝑦)

Proof of Theorem cdleme32e
StepHypRef Expression
1 simp23l 1292 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑃𝑄)
21pm2.24d 151 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (¬ 𝑃𝑄𝑋 (𝑁 (𝑌 𝑊))))
3 simp11l 1282 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝐾 ∈ HL)
43hllatd 38773 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝐾 ∈ Lat)
5 simp21l 1288 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋𝐵)
6 simp11r 1283 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑊𝐻)
7 cdleme32.b . . . . . . 7 𝐵 = (Base‘𝐾)
8 cdleme32.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
97, 8lhpbase 39408 . . . . . 6 (𝑊𝐻𝑊𝐵)
106, 9syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑊𝐵)
11 cdleme32.l . . . . . 6 = (le‘𝐾)
12 cdleme32.m . . . . . 6 = (meet‘𝐾)
137, 11, 12latleeqm1 18450 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊 ↔ (𝑋 𝑊) = 𝑋))
144, 5, 10, 13syl3anc 1369 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊 ↔ (𝑋 𝑊) = 𝑋))
157, 12latmcl 18423 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
164, 5, 10, 15syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) ∈ 𝐵)
17 simp21r 1289 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑌𝐵)
187, 12latmcl 18423 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
194, 17, 10, 18syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑌 𝑊) ∈ 𝐵)
20 simp11 1201 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
21 simp12 1202 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
22 simp13 1203 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
23 simp31 1207 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
24 cdleme32.j . . . . . . . . 9 = (join‘𝐾)
25 cdleme32.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
26 cdleme32.u . . . . . . . . 9 𝑈 = ((𝑃 𝑄) 𝑊)
27 cdleme32.c . . . . . . . . 9 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
28 cdleme32.d . . . . . . . . 9 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
29 cdleme32.e . . . . . . . . 9 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
30 cdleme32.i . . . . . . . . 9 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
31 cdleme32.n . . . . . . . . 9 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
327, 11, 24, 12, 25, 8, 26, 27, 28, 29, 30, 31cdleme27cl 39776 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝑁𝐵)
3320, 21, 22, 23, 1, 32syl122anc 1377 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑁𝐵)
347, 24latjcl 18422 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑁𝐵 ∧ (𝑌 𝑊) ∈ 𝐵) → (𝑁 (𝑌 𝑊)) ∈ 𝐵)
354, 33, 19, 34syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑁 (𝑌 𝑊)) ∈ 𝐵)
36 simp33 1209 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋 𝑌)
377, 11, 12latmlem1 18452 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑊𝐵)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
384, 5, 17, 10, 37syl13anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
3936, 38mpd 15 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) (𝑌 𝑊))
407, 11, 24latlej2 18432 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑁𝐵 ∧ (𝑌 𝑊) ∈ 𝐵) → (𝑌 𝑊) (𝑁 (𝑌 𝑊)))
414, 33, 19, 40syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑌 𝑊) (𝑁 (𝑌 𝑊)))
427, 11, 4, 16, 19, 35, 39, 41lattrd 18429 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) (𝑁 (𝑌 𝑊)))
43 breq1 5145 . . . . 5 ((𝑋 𝑊) = 𝑋 → ((𝑋 𝑊) (𝑁 (𝑌 𝑊)) ↔ 𝑋 (𝑁 (𝑌 𝑊))))
4442, 43syl5ibcom 244 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝑋 𝑊) = 𝑋𝑋 (𝑁 (𝑌 𝑊))))
4514, 44sylbid 239 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊𝑋 (𝑁 (𝑌 𝑊))))
46 simp22 1205 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊))
47 pm4.53 984 . . . 4 (¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ↔ (¬ 𝑃𝑄𝑋 𝑊))
4846, 47sylib 217 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (¬ 𝑃𝑄𝑋 𝑊))
492, 45, 48mpjaod 859 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋 (𝑁 (𝑌 𝑊)))
50 cdleme32.f . . . 4 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
5150cdleme31fv2 39803 . . 3 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
525, 46, 51syl2anc 583 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐹𝑋) = 𝑋)
53 simp1 1134 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
54 simp23 1206 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑃𝑄 ∧ ¬ 𝑌 𝑊))
55 simp32 1208 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑠 (𝑌 𝑊)) = 𝑌)
56 cdleme32.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
577, 11, 24, 12, 25, 8, 26, 27, 28, 29, 30, 31, 56, 50cdleme32a 39851 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌)) → (𝐹𝑌) = (𝑁 (𝑌 𝑊)))
5853, 17, 54, 23, 55, 57syl122anc 1377 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐹𝑌) = (𝑁 (𝑌 𝑊)))
5949, 52, 583brtr4d 5174 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  ifcif 4524   class class class wbr 5142  cmpt 5225  cfv 6542  crio 7369  (class class class)co 7414  Basecbs 17171  lecple 17231  joincjn 18294  meetcmee 18295  Latclat 18414  Atomscatm 38672  HLchlt 38759  LHypclh 39394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-riotaBAD 38362
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-undef 8272  df-proset 18278  df-poset 18296  df-plt 18313  df-lub 18329  df-glb 18330  df-join 18331  df-meet 18332  df-p0 18408  df-p1 18409  df-lat 18415  df-clat 18482  df-oposet 38585  df-ol 38587  df-oml 38588  df-covers 38675  df-ats 38676  df-atl 38707  df-cvlat 38731  df-hlat 38760  df-llines 38908  df-lplanes 38909  df-lvols 38910  df-lines 38911  df-psubsp 38913  df-pmap 38914  df-padd 39206  df-lhyp 39398
This theorem is referenced by:  cdleme32f  39856
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