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Theorem cdlemg33e 35824
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 1060 . . 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 1093 . . 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 1181 . . 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 1062 . . 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 35816 . . 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 1330 . 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 1171 . . . . . . . . . 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 34473 . . . . . . . . . 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 2621 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
1918, 8atn0 34421 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ 𝑧𝐴) → 𝑧 ≠ (0.‘𝐾))
2017, 19sylan 488 . . . . . . . 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 1179 . . . . . . . . 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 481 . . . . . . . 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 2867 . . . . . . 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 1180 . . . . . . . . 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 481 . . . . . . . 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 2867 . . . . . . 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 554 . . . . . 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 529 . . . . 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 1039 . . . . 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 740 . . 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 3047 . 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 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793  wrex 2912   class class class wbr 4651  cfv 5886  (class class class)co 6647  lecple 15942  joincjn 16938  meetcmee 16939  0.cp0 17031  Atomscatm 34376  AtLatcal 34377  HLchlt 34463  LHypclh 35096  LTrncltrn 35213  trLctrl 35271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-preset 16922  df-poset 16940  df-plt 16952  df-lub 16968  df-glb 16969  df-join 16970  df-meet 16971  df-p0 17033  df-p1 17034  df-lat 17040  df-clat 17102  df-oposet 34289  df-ol 34291  df-oml 34292  df-covers 34379  df-ats 34380  df-atl 34411  df-cvlat 34435  df-hlat 34464  df-llines 34610  df-lplanes 34611  df-lhyp 35100
This theorem is referenced by:  cdlemg33  35825
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