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Theorem cdlemefrs29cpre1 37420
Description: TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)
Hypotheses
Ref Expression
cdlemefrs27.b 𝐵 = (Base‘𝐾)
cdlemefrs27.l = (le‘𝐾)
cdlemefrs27.j = (join‘𝐾)
cdlemefrs27.m = (meet‘𝐾)
cdlemefrs27.a 𝐴 = (Atoms‘𝐾)
cdlemefrs27.h 𝐻 = (LHyp‘𝐾)
cdlemefrs27.eq (𝑠 = 𝑅 → (𝜑𝜓))
cdlemefrs27.nb ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
cdlemefrs27.rnb ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑠𝑁𝐵)
Assertion
Ref Expression
cdlemefrs29cpre1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
Distinct variable groups:   𝑧,𝑠,𝐴   𝐻,𝑠   ,𝑠   𝐾,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑊,𝑠   𝜓,𝑠   𝑧,𝐴   𝑧,𝐵   𝑧,𝐻   𝑧,𝐾   𝑧,   𝑧,𝑁   𝑧,𝑃   𝑧,𝑄   𝑧,𝑅   𝑧,𝑊   𝜓,𝑧   𝐵,𝑠   𝑧,   ,𝑠,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑠)   𝑁(𝑠)

Proof of Theorem cdlemefrs29cpre1
StepHypRef Expression
1 cdlemefrs27.b . . 3 𝐵 = (Base‘𝐾)
2 cdlemefrs27.l . . 3 = (le‘𝐾)
3 cdlemefrs27.j . . 3 = (join‘𝐾)
4 cdlemefrs27.m . . 3 = (meet‘𝐾)
5 cdlemefrs27.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdlemefrs27.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdlemefrs27.eq . . 3 (𝑠 = 𝑅 → (𝜑𝜓))
8 cdlemefrs27.nb . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄 ∧ (𝑠𝐴 ∧ (¬ 𝑠 𝑊𝜑))) → 𝑁𝐵)
9 cdlemefrs27.rnb . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅 / 𝑠𝑁𝐵)
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemefrs29bpre1 37419 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
11 simp11 1197 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp2rl 1236 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅𝐴)
131, 5atbase 36311 . . . . . 6 (𝑅𝐴𝑅𝐵)
1412, 13syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝑅𝐵)
15 simp2rr 1237 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ¬ 𝑅 𝑊)
161, 2, 3, 4, 5, 6lhpmcvr2 37046 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐵 ∧ ¬ 𝑅 𝑊)) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅))
1711, 14, 15, 16syl12anc 834 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅))
18 simpl3 1187 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → 𝜓)
197pm5.32ri 576 . . . . . . . . . 10 ((𝜑𝑠 = 𝑅) ↔ (𝜓𝑠 = 𝑅))
2019baibr 537 . . . . . . . . 9 (𝜓 → (𝑠 = 𝑅 ↔ (𝜑𝑠 = 𝑅)))
2118, 20syl 17 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑠 = 𝑅 ↔ (𝜑𝑠 = 𝑅)))
22 simp2r 1194 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
23 eqid 2826 . . . . . . . . . . . . . 14 (0.‘𝐾) = (0.‘𝐾)
242, 4, 23, 5, 6lhpmat 37052 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (0.‘𝐾))
2511, 22, 24syl2anc 584 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (𝑅 𝑊) = (0.‘𝐾))
2625adantr 481 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑅 𝑊) = (0.‘𝐾))
2726oveq2d 7166 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑠 (𝑅 𝑊)) = (𝑠 (0.‘𝐾)))
28 simp11l 1278 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐾 ∈ HL)
29 hlol 36383 . . . . . . . . . . . 12 (𝐾 ∈ HL → 𝐾 ∈ OL)
3028, 29syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → 𝐾 ∈ OL)
311, 5atbase 36311 . . . . . . . . . . 11 (𝑠𝐴𝑠𝐵)
321, 3, 23olj01 36247 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑠𝐵) → (𝑠 (0.‘𝐾)) = 𝑠)
3330, 31, 32syl2an 595 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑠 (0.‘𝐾)) = 𝑠)
3427, 33eqtrd 2861 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → (𝑠 (𝑅 𝑊)) = 𝑠)
3534eqeq1d 2828 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → ((𝑠 (𝑅 𝑊)) = 𝑅𝑠 = 𝑅))
3635anbi2d 628 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → ((𝜑 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (𝜑𝑠 = 𝑅)))
3721, 35, 363bitr4d 312 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → ((𝑠 (𝑅 𝑊)) = 𝑅 ↔ (𝜑 ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
3837anbi2d 628 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (¬ 𝑠 𝑊 ∧ (𝜑 ∧ (𝑠 (𝑅 𝑊)) = 𝑅))))
39 anass 469 . . . . . 6 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (¬ 𝑠 𝑊 ∧ (𝜑 ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
4038, 39syl6bbr 290 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) ∧ 𝑠𝐴) → ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
4140rexbidva 3301 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∃𝑠𝐴𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ ∃𝑠𝐴 ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))
4217, 41mpbid 233 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃𝑠𝐴 ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅))
43 reusv1 5294 . . 3 (∃𝑠𝐴 ((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → (∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ∃𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
4442, 43syl 17 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → (∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ ∃𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊)))))
4510, 44mpbird 258 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝜓) → ∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝜑) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  ∃!wreu 3145  csb 3887   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  joincjn 17549  meetcmee 17550  0.cp0 17642  OLcol 36196  Atomscatm 36285  HLchlt 36372  LHypclh 37006
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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-lhyp 37010
This theorem is referenced by:  cdlemefrs29clN  37421  cdlemefrs32fva  37422  cdlemefs29cpre1N  37440
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