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Theorem cdlemg33e 36512
Description: TODO: Fix comment. (Contributed by NM, 30-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l = (le‘𝐾)
cdlemg12.j = (join‘𝐾)
cdlemg12.m = (meet‘𝐾)
cdlemg12.a 𝐴 = (Atoms‘𝐾)
cdlemg12.h 𝐻 = (LHyp‘𝐾)
cdlemg12.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg31.n 𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))
cdlemg33.o 𝑂 = ((𝑃 𝑣) (𝑄 (𝑅𝐺)))
Assertion
Ref Expression
cdlemg33e ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣))))
Distinct variable groups:   𝐴,𝑟   𝐺,𝑟   ,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑊,𝑟   𝐹,𝑟   𝑧,𝐴   𝑧,𝐹,𝑟   𝐻,𝑟,𝑧   𝑧,   𝐾,𝑟,𝑧   𝑧,   𝑁,𝑟,𝑧   𝑧,𝑃   𝑧,𝑄   𝑧,𝑅   𝑧,𝑇   𝑧,𝑊   𝑧,𝑣,𝑟   𝑧,𝐺   𝑧,𝑂,𝑟
Allowed substitution hints:   𝐴(𝑣)   𝑃(𝑣)   𝑄(𝑣)   𝑅(𝑣,𝑟)   𝑇(𝑣,𝑟)   𝐹(𝑣)   𝐺(𝑣)   𝐻(𝑣)   (𝑣)   𝐾(𝑣)   (𝑣)   (𝑧,𝑣,𝑟)   𝑁(𝑣)   𝑂(𝑣)   𝑊(𝑣)

Proof of Theorem cdlemg33e
StepHypRef Expression
1 simp1 1129 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
2 simp21 1247 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑣𝐴𝑣 𝑊))
3 simp23l 1377 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐹𝑇)
4 simp3 1131 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟))))
5 cdlemg12.l . . . 4 = (le‘𝐾)
6 cdlemg12.j . . . 4 = (join‘𝐾)
7 cdlemg12.m . . . 4 = (meet‘𝐾)
8 cdlemg12.a . . . 4 𝐴 = (Atoms‘𝐾)
9 cdlemg12.h . . . 4 𝐻 = (LHyp‘𝐾)
10 cdlemg12.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
11 cdlemg12b.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
12 cdlemg31.n . . . 4 𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))
135, 6, 7, 8, 9, 10, 11, 12cdlemg33c0 36504 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ 𝐹𝑇) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊𝑧 (𝑃 𝑣)))
141, 2, 3, 4, 13syl121anc 1480 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊𝑧 (𝑃 𝑣)))
15 simp11l 1367 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
16 hlatl 35162 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1715, 16syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ AtLat)
18 eqid 2770 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
1918, 8atn0 35110 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑧𝐴) → 𝑧 ≠ (0.‘𝐾))
2017, 19sylan 561 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → 𝑧 ≠ (0.‘𝐾))
21 simp22l 1375 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑁 = (0.‘𝐾))
2221adantr 466 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → 𝑁 = (0.‘𝐾))
2320, 22neeqtrrd 3016 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → 𝑧𝑁)
24 simp22r 1376 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑂 = (0.‘𝐾))
2524adantr 466 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → 𝑂 = (0.‘𝐾))
2620, 25neeqtrrd 3016 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → 𝑧𝑂)
2723, 26jca 495 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → (𝑧𝑁𝑧𝑂))
2827biantrurd 516 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → (𝑧 (𝑃 𝑣) ↔ ((𝑧𝑁𝑧𝑂) ∧ 𝑧 (𝑃 𝑣))))
29 df-3an 1072 . . . . 5 ((𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣)) ↔ ((𝑧𝑁𝑧𝑂) ∧ 𝑧 (𝑃 𝑣)))
3028, 29syl6bbr 278 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → (𝑧 (𝑃 𝑣) ↔ (𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣))))
3130anbi2d 606 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑧𝐴) → ((¬ 𝑧 𝑊𝑧 (𝑃 𝑣)) ↔ (¬ 𝑧 𝑊 ∧ (𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣)))))
3231rexbidva 3196 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (∃𝑧𝐴𝑧 𝑊𝑧 (𝑃 𝑣)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣)))))
3314, 32mpbid 222 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹𝑇𝐺𝑇)) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wrex 3061   class class class wbr 4784  cfv 6031  (class class class)co 6792  lecple 16155  joincjn 17151  meetcmee 17152  0.cp0 17244  Atomscatm 35065  AtLatcal 35066  HLchlt 35152  LHypclh 35785  LTrncltrn 35902  trLctrl 35960
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
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  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-ral 3065  df-rex 3066  df-reu 3067  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-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-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-lhyp 35789
This theorem is referenced by:  cdlemg33  36513
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