Theorem cdlemg33 37240

Description: Combine cdlemg33b 37236, cdlemg33c 37237, cdlemg33d 37238, cdlemg33e 37239. 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 1183	. . 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 1184	. . 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 1185	. . 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 1186	. . 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 1272	. . 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 1189	. . 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 37224	. . 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 1368	. 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 1273	. . 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 1190	. . 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 37224	. . 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 1368	. 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 1171	. . . . 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 1231	. . . . 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 477	. . . . 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 1232	. . . . 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 1233	. . . . 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 1234	. . . . 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 1236	. . . . 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 37236	. . . . 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 1373	. . . 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 405	. . 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 1171	. . . . 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 1231	. . . . 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 477	. . . . 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 1232	. . . . 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 1233	. . . . 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 1235	. . . . 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 1236	. . . . 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 37238	. . . . 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 1373	. . . 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 405	. . 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 1171	. . . . 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 1231	. . . . 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 477	. . . . 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 1232	. . . . 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 1233	. . . . 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 1234	. . . . 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 1236	. . . . 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 37237	. . . . 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 1373	. . . 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 405	. . 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 1171	. . . . 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 1231	. . . . 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 477	. . . . 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 1232	. . . . 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 1233	. . . . 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 1234	. . . . 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 1236	. . . . 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 37239	. . . . 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 1373	. . . 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 405	. . 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 1019	. 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 686	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 387   ∨ wo 833   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048   ≠ wne 2961  ∃wrex 3083   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970  lecple 16418  joincjn 17402  meetcmee 17403  0.cp0 17495  Atomscatm 35792  HLchlt 35879  LHypclh 36513  LTrncltrn 36630  trLctrl 36687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-map 8200  df-proset 17386  df-poset 17404  df-plt 17416  df-lub 17432  df-glb 17433  df-join 17434  df-meet 17435  df-p0 17497  df-p1 17498  df-lat 17504  df-clat 17566  df-oposet 35705  df-ol 35707  df-oml 35708  df-covers 35795  df-ats 35796  df-atl 35827  df-cvlat 35851  df-hlat 35880  df-llines 36027  df-lplanes 36028  df-psubsp 36032  df-pmap 36033  df-padd 36325  df-lhyp 36517  df-laut 36518  df-ldil 36633  df-ltrn 36634  df-trl 36688

This theorem is referenced by:  cdlemg34  37241