Theorem cdlemg33 41148

Description: Combine cdlemg33b 41144, cdlemg33c 41145, cdlemg33d 41146, cdlemg33e 41147. 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 ∨ (𝑅‘𝐹)))
cdlemg33.o	⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))

Assertion

Ref	Expression
cdlemg33	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (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 cdlemg33

Step	Hyp	Ref	Expression
1		simp11 1205	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp12 1206	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simp13 1207	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simp21 1208	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
5		simp22l 1294	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐹 ∈ 𝑇)
6		simp31 1211	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑣 ≠ (𝑅‘𝐹))
7		cdlemg12.l	. . . 4 ⊢ ≤ = (le‘𝐾)
8		cdlemg12.j	. . . 4 ⊢ ∨ = (join‘𝐾)
9		cdlemg12.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
10		cdlemg12.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemg12.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdlemg12.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13		cdlemg12b.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14		cdlemg31.n	. . . 4 ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))
15	7, 8, 9, 10, 11, 12, 13, 14	cdlemg31b0a 41132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹))) → (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)))
16	1, 2, 3, 4, 5, 6, 15	syl132anc 1391	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)))
17		simp22r 1295	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐺 ∈ 𝑇)
18		simp32 1212	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑣 ≠ (𝑅‘𝐺))
19		cdlemg33.o	. . . 4 ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))
20	7, 8, 9, 10, 11, 12, 13, 19	cdlemg31b0a 41132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐺))) → (𝑂 ∈ 𝐴 ∨ 𝑂 = (0.‘𝐾)))
21	1, 2, 3, 4, 17, 18, 20	syl132anc 1391	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑂 ∈ 𝐴 ∨ 𝑂 = (0.‘𝐾)))
22		simpl1 1193	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
23		simpl21 1253	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
24		simpr 484	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴))
25		simpl22 1254	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
26		simpl23 1255	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → 𝑃 ≠ 𝑄)
27		simpl31 1256	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → 𝑣 ≠ (𝑅‘𝐹))
28		simpl33 1258	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
29	7, 8, 9, 10, 11, 12, 13, 14, 19	cdlemg33b 41144	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
30	22, 23, 24, 25, 26, 27, 28, 29	syl133anc 1396	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
31	30	ex 412	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)))))
32		simpl1 1193	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
33		simpl21 1253	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
34		simpr 484	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴))
35		simpl22 1254	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
36		simpl23 1255	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → 𝑃 ≠ 𝑄)
37		simpl32 1257	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → 𝑣 ≠ (𝑅‘𝐺))
38		simpl33 1258	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
39	7, 8, 9, 10, 11, 12, 13, 14, 19	cdlemg33d 41146	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
40	32, 33, 34, 35, 36, 37, 38, 39	syl133anc 1396	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
41	40	ex 412	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)))))
42		simpl1 1193	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
43		simpl21 1253	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
44		simpr 484	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾)))
45		simpl22 1254	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
46		simpl23 1255	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → 𝑃 ≠ 𝑄)
47		simpl31 1256	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → 𝑣 ≠ (𝑅‘𝐹))
48		simpl33 1258	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
49	7, 8, 9, 10, 11, 12, 13, 14, 19	cdlemg33c 41145	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
50	42, 43, 44, 45, 46, 47, 48, 49	syl133anc 1396	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
51	50	ex 412	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)))))
52		simpl1 1193	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
53		simpl21 1253	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
54		simpr 484	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)))
55		simpl22 1254	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
56		simpl23 1255	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → 𝑃 ≠ 𝑄)
57		simpl31 1256	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → 𝑣 ≠ (𝑅‘𝐹))
58		simpl33 1258	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
59	7, 8, 9, 10, 11, 12, 13, 14, 19	cdlemg33e 41147	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
60	52, 53, 54, 55, 56, 57, 58, 59	syl133anc 1396	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
61	60	ex 412	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)))))
62	31, 41, 51, 61	ccased 1039	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (((𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)) ∧ (𝑂 ∈ 𝐴 ∨ 𝑂 = (0.‘𝐾))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)))))
63	16, 21, 62	mp2and 700	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   class class class wbr 5086  ‘cfv 6490  (class class class)co 7358  lecple 17185  joincjn 18235  meetcmee 18236  0.cp0 18345  Atomscatm 39700  HLchlt 39787  LHypclh 40421  LTrncltrn 40538  trLctrl 40595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8766  df-proset 18218  df-poset 18237  df-plt 18252  df-lub 18268  df-glb 18269  df-join 18270  df-meet 18271  df-p0 18347  df-p1 18348  df-lat 18356  df-clat 18423  df-oposet 39613  df-ol 39615  df-oml 39616  df-covers 39703  df-ats 39704  df-atl 39735  df-cvlat 39759  df-hlat 39788  df-llines 39935  df-lplanes 39936  df-psubsp 39940  df-pmap 39941  df-padd 40233  df-lhyp 40425  df-laut 40426  df-ldil 40541  df-ltrn 40542  df-trl 40596

This theorem is referenced by:  cdlemg34  41149