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Theorem cdleme32fva 37555
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 1193 . . . 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 36407 . . . 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 2819 . . . 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 37497 . . 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 1130 . . . 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 1132 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑃𝑄)
12 simp2 1131 . . . 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 37554 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 / 𝑠𝑁𝐵)
2410, 11, 12, 23syl12anc 834 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑠𝑁𝐵)
25 nfv 1908 . . . . . . . . 9 𝑠 ¬ 𝑅 𝑊
26 nfcsb1v 3905 . . . . . . . . . 10 𝑠𝑅 / 𝑠𝑁
2726nfeq2 2993 . . . . . . . . 9 𝑠 𝑧 = 𝑅 / 𝑠𝑁
2825, 27nfim 1890 . . . . . . . 8 𝑠𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)
29 breq1 5060 . . . . . . . . . . 11 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
3029notbid 320 . . . . . . . . . 10 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
31 csbeq1a 3895 . . . . . . . . . . 11 (𝑠 = 𝑅𝑁 = 𝑅 / 𝑠𝑁)
3231eqeq2d 2830 . . . . . . . . . 10 (𝑠 = 𝑅 → (𝑧 = 𝑁𝑧 = 𝑅 / 𝑠𝑁))
3330, 32imbi12d 347 . . . . . . . . 9 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3433ax-gen 1789 . . . . . . . 8 𝑠(𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
35 ceqsralt 3527 . . . . . . . 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 1444 . . . . . . 7 (𝑅𝐴 → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3736adantr 483 . . . . . 6 ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
38373ad2ant2 1128 . . . . 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 1197 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
40 eqid 2819 . . . . . . . . . . . . . . . 16 (0.‘𝐾) = (0.‘𝐾)
4113, 15, 40, 3, 16lhpmat 37148 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
4239, 12, 41syl2anc 586 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝑅 𝑊) = (0.‘𝐾))
4342adantr 483 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
4443oveq2d 7164 . . . . . . . . . . . 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 1278 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4645adantr 483 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ HL)
47 hlol 36479 . . . . . . . . . . . . . 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 36407 . . . . . . . . . . . . . 14 (𝑠𝐴𝑠𝐵)
5049ad2antrl 726 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑠𝐵)
512, 14, 40olj01 36343 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
5248, 50, 51syl2anc 586 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (0.‘𝐾)) = 𝑠)
5344, 52eqtrd 2854 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (𝑅 𝑊)) = 𝑠)
5453eqeq1d 2821 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
5543oveq2d 7164 . . . . . . . . . . . 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 1242 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
57 simpl12 1243 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
58 simpl13 1244 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
59 simpr 487 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
60 simpl3 1187 . . . . . . . . . . . . . 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 37484 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝑁𝐵)
6256, 57, 58, 59, 60, 61syl122anc 1373 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑁𝐵)
632, 14, 40olj01 36343 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑁𝐵) → (𝑁 (0.‘𝐾)) = 𝑁)
6448, 62, 63syl2anc 586 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (0.‘𝐾)) = 𝑁)
6555, 64eqtrd 2854 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (𝑅 𝑊)) = 𝑁)
6665eqeq2d 2830 . . . . . . . . . 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 347 . . . . . . . . 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 459 . . . . . . . 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 275 . . . . . . 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 453 . . . . . . 7 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ (¬ 𝑠 𝑊 → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
71 bi2.04 391 . . . . . . 7 ((𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑠 𝑊 → (𝑠 = 𝑅𝑧 = 𝑁)))
7269, 70, 713bitr4g 316 . . . . . 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 3194 . . . . 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 1194 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → ¬ 𝑅 𝑊)
75 biimt 363 . . . . . 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 313 . . . 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 483 . . 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 7135 . 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 2854 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 208  wa 398  w3a 1081  wal 1528   = wceq 1530  wnf 1777  wcel 2107  wne 3014  wral 3136  csb 3881  ifcif 4465   class class class wbr 5057  cmpt 5137  cfv 6348  crio 7105  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  0.cp0 17639  OLcol 36292  Atomscatm 36381  HLchlt 36468  LHypclh 37102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36071
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-undef 7931  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36294  df-ol 36296  df-oml 36297  df-covers 36384  df-ats 36385  df-atl 36416  df-cvlat 36440  df-hlat 36469  df-llines 36616  df-lplanes 36617  df-lvols 36618  df-lines 36619  df-psubsp 36621  df-pmap 36622  df-padd 36914  df-lhyp 37106
This theorem is referenced by:  cdleme32fva1  37556
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