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Theorem cdleme32fva 40439
Description: Part of proof of Lemma D in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. (Contributed by NM, 2-Mar-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
cdleme32fva ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)
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   𝑅,𝑠,𝑡,𝑦   𝑦,𝐻   𝑦,𝐾   𝑥,𝑅,𝑧   𝑧,𝐻   𝑧,𝐾
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)

Proof of Theorem cdleme32fva
StepHypRef Expression
1 simp2l 1200 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅𝐴)
2 cdleme32.b . . . . 5 𝐵 = (Base‘𝐾)
3 cdleme32.a . . . . 5 𝐴 = (Atoms‘𝐾)
42, 3atbase 39290 . . . 4 (𝑅𝐴𝑅𝐵)
51, 4syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅𝐵)
6 cdleme32.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
7 eqid 2737 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
86, 7cdleme31so 40381 . . 3 (𝑅𝐵𝑅 / 𝑥𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
95, 8syl 17 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑥𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
10 simp1 1137 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
11 simp3 1139 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑃𝑄)
12 simp2 1138 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
13 cdleme32.l . . . . 5 = (le‘𝐾)
14 cdleme32.j . . . . 5 = (join‘𝐾)
15 cdleme32.m . . . . 5 = (meet‘𝐾)
16 cdleme32.h . . . . 5 𝐻 = (LHyp‘𝐾)
17 cdleme32.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
18 cdleme32.c . . . . 5 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
19 cdleme32.d . . . . 5 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
20 cdleme32.e . . . . 5 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
21 cdleme32.i . . . . 5 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))
22 cdleme32.n . . . . 5 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
232, 13, 14, 15, 3, 16, 17, 18, 19, 20, 21, 22cdleme32snb 40438 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 / 𝑠𝑁𝐵)
2410, 11, 12, 23syl12anc 837 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑠𝑁𝐵)
25 nfv 1914 . . . . . . . . 9 𝑠 ¬ 𝑅 𝑊
26 nfcsb1v 3923 . . . . . . . . . 10 𝑠𝑅 / 𝑠𝑁
2726nfeq2 2923 . . . . . . . . 9 𝑠 𝑧 = 𝑅 / 𝑠𝑁
2825, 27nfim 1896 . . . . . . . 8 𝑠𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)
29 breq1 5146 . . . . . . . . . . 11 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
3029notbid 318 . . . . . . . . . 10 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
31 csbeq1a 3913 . . . . . . . . . . 11 (𝑠 = 𝑅𝑁 = 𝑅 / 𝑠𝑁)
3231eqeq2d 2748 . . . . . . . . . 10 (𝑠 = 𝑅 → (𝑧 = 𝑁𝑧 = 𝑅 / 𝑠𝑁))
3330, 32imbi12d 344 . . . . . . . . 9 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3433ax-gen 1795 . . . . . . . 8 𝑠(𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
35 ceqsralt 3516 . . . . . . . 8 ((Ⅎ𝑠𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁) ∧ ∀𝑠(𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁))) ∧ 𝑅𝐴) → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3628, 34, 35mp3an12 1453 . . . . . . 7 (𝑅𝐴 → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3736adantr 480 . . . . . 6 ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
38373ad2ant2 1135 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
39 simp11 1204 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
40 eqid 2737 . . . . . . . . . . . . . . . 16 (0.‘𝐾) = (0.‘𝐾)
4113, 15, 40, 3, 16lhpmat 40032 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
4239, 12, 41syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝑅 𝑊) = (0.‘𝐾))
4342adantr 480 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
4443oveq2d 7447 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (𝑅 𝑊)) = (𝑠 (0.‘𝐾)))
45 simp11l 1285 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4645adantr 480 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ HL)
47 hlol 39362 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → 𝐾 ∈ OL)
4846, 47syl 17 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ OL)
492, 3atbase 39290 . . . . . . . . . . . . . 14 (𝑠𝐴𝑠𝐵)
5049ad2antrl 728 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑠𝐵)
512, 14, 40olj01 39226 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
5248, 50, 51syl2anc 584 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (0.‘𝐾)) = 𝑠)
5344, 52eqtrd 2777 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (𝑅 𝑊)) = 𝑠)
5453eqeq1d 2739 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
5543oveq2d 7447 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (𝑅 𝑊)) = (𝑁 (0.‘𝐾)))
56 simpl11 1249 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
57 simpl12 1250 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
58 simpl13 1251 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
59 simpr 484 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
60 simpl3 1194 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑃𝑄)
612, 13, 14, 15, 3, 16, 17, 18, 19, 20, 21, 22cdleme27cl 40368 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝑁𝐵)
6256, 57, 58, 59, 60, 61syl122anc 1381 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑁𝐵)
632, 14, 40olj01 39226 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑁𝐵) → (𝑁 (0.‘𝐾)) = 𝑁)
6448, 62, 63syl2anc 584 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (0.‘𝐾)) = 𝑁)
6555, 64eqtrd 2777 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (𝑅 𝑊)) = 𝑁)
6665eqeq2d 2748 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑧 = (𝑁 (𝑅 𝑊)) ↔ 𝑧 = 𝑁))
6754, 66imbi12d 344 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊))) ↔ (𝑠 = 𝑅𝑧 = 𝑁)))
6867expr 456 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ 𝑠𝐴) → (¬ 𝑠 𝑊 → (((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊))) ↔ (𝑠 = 𝑅𝑧 = 𝑁))))
6968pm5.74d 273 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ 𝑠𝐴) → ((¬ 𝑠 𝑊 → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))) ↔ (¬ 𝑠 𝑊 → (𝑠 = 𝑅𝑧 = 𝑁))))
70 impexp 450 . . . . . . 7 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ (¬ 𝑠 𝑊 → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
71 bi2.04 387 . . . . . . 7 ((𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑠 𝑊 → (𝑠 = 𝑅𝑧 = 𝑁)))
7269, 70, 713bitr4g 314 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ 𝑠𝐴) → (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁))))
7372ralbidva 3176 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁))))
74 simp2r 1201 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → ¬ 𝑅 𝑊)
75 biimt 360 . . . . . 6 𝑅 𝑊 → (𝑧 = 𝑅 / 𝑠𝑁 ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
7674, 75syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝑧 = 𝑅 / 𝑠𝑁 ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
7738, 73, 763bitr4d 311 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
7877adantr 480 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ 𝑧𝐵) → (∀𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ 𝑧 = 𝑅 / 𝑠𝑁))
7924, 78riota5 7417 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))) = 𝑅 / 𝑠𝑁)
809, 79eqtrd 2777 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wnf 1783  wcel 2108  wne 2940  wral 3061  csb 3899  ifcif 4525   class class class wbr 5143  cmpt 5225  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  OLcol 39175  Atomscatm 39264  HLchlt 39351  LHypclh 39986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-undef 8298  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990
This theorem is referenced by:  cdleme32fva1  40440
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