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Theorem cdleme32le 34552
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
cdleme32le ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 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 cdleme32le
StepHypRef Expression
1 simpl1 1056 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
2 simpl2l 1106 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
3 simpl2r 1107 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑌𝐵)
4 simpr 475 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝑃𝑄 ∧ ¬ 𝑋 𝑊))
5 simpl3 1058 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋 𝑌)
6 cdleme32.b . . . 4 𝐵 = (Base‘𝐾)
7 cdleme32.l . . . 4 = (le‘𝐾)
8 cdleme32.j . . . 4 = (join‘𝐾)
9 cdleme32.m . . . 4 = (meet‘𝐾)
10 cdleme32.a . . . 4 𝐴 = (Atoms‘𝐾)
11 cdleme32.h . . . 4 𝐻 = (LHyp‘𝐾)
12 cdleme32.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
13 cdleme32.c . . . 4 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
14 cdleme32.d . . . 4 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
15 cdleme32.e . . . 4 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
16 cdleme32.i . . . 4 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
17 cdleme32.n . . . 4 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
18 cdleme32.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
19 cdleme32.f . . . 4 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
206, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme32d 34549 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))
211, 2, 3, 4, 5, 20syl131anc 1330 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) (𝐹𝑌))
22 simp11 1083 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
23 simp12 1084 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝑋𝐵𝑌𝐵))
24 simp3 1055 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊))
25 simp2 1054 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝑃𝑄 ∧ ¬ 𝑌 𝑊))
26 simp13 1085 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋 𝑌)
276, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme32f 34551 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))
2822, 23, 24, 25, 26, 27syl131anc 1330 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) (𝐹𝑌))
29283exp 1255 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → ((𝑃𝑄 ∧ ¬ 𝑌 𝑊) → (¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) → (𝐹𝑋) (𝐹𝑌))))
30 simp13 1085 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋 𝑌)
31 simp12l 1166 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
32 simp3 1055 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊))
3319cdleme31fv2 34498 . . . . . . 7 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
3431, 32, 33syl2anc 690 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
35 simp12r 1167 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑌𝐵)
36 simp2 1054 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊))
3719cdleme31fv2 34498 . . . . . . 7 ((𝑌𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) → (𝐹𝑌) = 𝑌)
3835, 36, 37syl2anc 690 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑌) = 𝑌)
3930, 34, 383brtr4d 4605 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) (𝐹𝑌))
40393exp 1255 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (¬ (𝑃𝑄 ∧ ¬ 𝑌 𝑊) → (¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) → (𝐹𝑋) (𝐹𝑌))))
4129, 40pm2.61d 168 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) → (𝐹𝑋) (𝐹𝑌)))
4241imp 443 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) (𝐹𝑌))
4321, 42pm2.61dan 827 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wral 2891  ifcif 4031   class class class wbr 4573  cmpt 4633  cfv 5786  crio 6484  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  Atomscatm 33367  HLchlt 33454  LHypclh 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-riotaBAD 33056
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-undef 7259  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-p1 16805  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-llines 33601  df-lplanes 33602  df-lvols 33603  df-lines 33604  df-psubsp 33606  df-pmap 33607  df-padd 33899  df-lhyp 34091
This theorem is referenced by:  cdlemeg49le  34616  cdlemeg49lebilem  34644
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