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Theorem cdleme32fva 36239
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 1240 . . . 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 35091 . . . 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 2770 . . . 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 36181 . . 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 1129 . . . 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 1131 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑃𝑄)
12 simp2 1130 . . . 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 36238 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 / 𝑠𝑁𝐵)
2410, 11, 12, 23syl12anc 1473 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑠𝑁𝐵)
25 nfv 1994 . . . . . . . . 9 𝑠 ¬ 𝑅 𝑊
26 nfcsb1v 3696 . . . . . . . . . 10 𝑠𝑅 / 𝑠𝑁
2726nfeq2 2928 . . . . . . . . 9 𝑠 𝑧 = 𝑅 / 𝑠𝑁
2825, 27nfim 1976 . . . . . . . 8 𝑠𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)
29 breq1 4787 . . . . . . . . . . 11 (𝑠 = 𝑅 → (𝑠 𝑊𝑅 𝑊))
3029notbid 307 . . . . . . . . . 10 (𝑠 = 𝑅 → (¬ 𝑠 𝑊 ↔ ¬ 𝑅 𝑊))
31 csbeq1a 3689 . . . . . . . . . . 11 (𝑠 = 𝑅𝑁 = 𝑅 / 𝑠𝑁)
3231eqeq2d 2780 . . . . . . . . . 10 (𝑠 = 𝑅 → (𝑧 = 𝑁𝑧 = 𝑅 / 𝑠𝑁))
3330, 32imbi12d 333 . . . . . . . . 9 (𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3433ax-gen 1869 . . . . . . . 8 𝑠(𝑠 = 𝑅 → ((¬ 𝑠 𝑊𝑧 = 𝑁) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
35 ceqsralt 3378 . . . . . . . 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 1561 . . . . . . 7 (𝑅𝐴 → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
3736adantr 466 . . . . . 6 ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) → (∀𝑠𝐴 (𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑅 𝑊𝑧 = 𝑅 / 𝑠𝑁)))
38373ad2ant2 1127 . . . . 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 1244 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
40 eqid 2770 . . . . . . . . . . . . . . . 16 (0.‘𝐾) = (0.‘𝐾)
4113, 15, 40, 3, 16lhpmat 35831 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
4239, 12, 41syl2anc 565 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝑅 𝑊) = (0.‘𝐾))
4342adantr 466 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
4443oveq2d 6808 . . . . . . . . . . . 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 1367 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4645adantr 466 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝐾 ∈ HL)
47 hlol 35163 . . . . . . . . . . . . . 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 35091 . . . . . . . . . . . . . 14 (𝑠𝐴𝑠𝐵)
5049ad2antrl 699 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑠𝐵)
512, 14, 40olj01 35027 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
5248, 50, 51syl2anc 565 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (0.‘𝐾)) = 𝑠)
5344, 52eqtrd 2804 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠 (𝑅 𝑊)) = 𝑠)
5453eqeq1d 2772 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
5543oveq2d 6808 . . . . . . . . . . . 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 1313 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
57 simpl12 1315 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
58 simpl13 1317 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
59 simpr 471 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑠𝐴 ∧ ¬ 𝑠 𝑊))
60 simpl3 1230 . . . . . . . . . . . . . 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 36168 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑃𝑄)) → 𝑁𝐵)
6256, 57, 58, 59, 60, 61syl122anc 1484 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → 𝑁𝐵)
632, 14, 40olj01 35027 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑁𝐵) → (𝑁 (0.‘𝐾)) = 𝑁)
6448, 62, 63syl2anc 565 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (0.‘𝐾)) = 𝑁)
6555, 64eqtrd 2804 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) ∧ (𝑠𝐴 ∧ ¬ 𝑠 𝑊)) → (𝑁 (𝑅 𝑊)) = 𝑁)
6665eqeq2d 2780 . . . . . . . . . 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 333 . . . . . . . . 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 444 . . . . . . . 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 262 . . . . . . 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 437 . . . . . . 7 (((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ (¬ 𝑠 𝑊 → ((𝑠 (𝑅 𝑊)) = 𝑅𝑧 = (𝑁 (𝑅 𝑊)))))
71 bi2.04 375 . . . . . . 7 ((𝑠 = 𝑅 → (¬ 𝑠 𝑊𝑧 = 𝑁)) ↔ (¬ 𝑠 𝑊 → (𝑠 = 𝑅𝑧 = 𝑁)))
7269, 70, 713bitr4g 303 . . . . . 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 3133 . . . . 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 1241 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → ¬ 𝑅 𝑊)
75 biimt 349 . . . . . 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 300 . . . 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 466 . . 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 6779 . 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 2804 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 196  wa 382  w3a 1070  wal 1628   = wceq 1630  wnf 1855  wcel 2144  wne 2942  wral 3060  csb 3680  ifcif 4223   class class class wbr 4784  cmpt 4861  cfv 6031  crio 6752  (class class class)co 6792  Basecbs 16063  lecple 16155  joincjn 17151  meetcmee 17152  0.cp0 17244  OLcol 34976  Atomscatm 35065  HLchlt 35152  LHypclh 35785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-undef 7550  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789
This theorem is referenced by:  cdleme32fva1  36240
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