Theorem 4atexlemex6 40076

Description: Lemma for 4atexlem7 40077. (Contributed by NM, 25-Nov-2012.)

Hypotheses

Ref	Expression
4thatleme.l	⊢ ≤ = (le‘𝐾)
4thatleme.j	⊢ ∨ = (join‘𝐾)
4thatleme.m	⊢ ∧ = (meet‘𝐾)
4thatleme.a	⊢ 𝐴 = (Atoms‘𝐾)
4thatleme.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
4atexlemex6	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Distinct variable groups:   𝑧,𝐴   𝑧, ∨   𝑧, ≤   𝑧, ∧   𝑧,𝑃   𝑧,𝑄   𝑧,𝑅   𝑧,𝑆   𝑧,𝑊

Allowed substitution hints:   𝐻(𝑧)   𝐾(𝑧)

Proof of Theorem 4atexlemex6

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11l 1285	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
2		simp11 1204	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp12 1205	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simp13l 1289	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
5		simp32 1211	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑄)
6		4thatleme.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
7		4thatleme.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
8		4thatleme.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
9		4thatleme.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
10		4thatleme.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
11	6, 7, 8, 9, 10	lhpat 40045	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴)
12	2, 3, 4, 5, 11	syl112anc 1376	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴)
13		simp2r 1201	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ 𝐴)
14		simp12l 1287	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
15		simp33 1212	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
16	6, 7, 9	atnlej1 39381	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ≠ 𝑃)
17	1, 13, 14, 4, 15, 16	syl131anc 1385	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ≠ 𝑃)
18	17	necomd 2996	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑆)
19	6, 7, 8, 9, 10	lhpat 40045	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴)
20	2, 3, 13, 18, 19	syl112anc 1376	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴)
21	7, 9	hlsupr2 39389	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴 ∧ ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴) → ∃𝑡 ∈ 𝐴 (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))
22	1, 12, 20, 21	syl3anc 1373	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ∃𝑡 ∈ 𝐴 (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))
23		simp111 1303	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24		simp112 1304	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
25		simp113 1305	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
26		simp12r 1288	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → 𝑆 ∈ 𝐴)
27		simp2ll 1241	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
28	27	3ad2ant1 1134	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → 𝑅 ∈ 𝐴)
29		simp2lr 1242	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑅 ≤ 𝑊)
30	29	3ad2ant1 1134	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → ¬ 𝑅 ≤ 𝑊)
31		simp131 1309	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
32	28, 30, 31	3jca 1129	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
33		3simpc 1151	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → (𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)))
34		simp132 1310	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → 𝑃 ≠ 𝑄)
35		simp133 1311	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
36		biid 261	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
37		eqid 2737	. . . . . 6 ⊢ ((𝑃 ∨ 𝑄) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)
38		eqid 2737	. . . . . 6 ⊢ ((𝑃 ∨ 𝑆) ∧ 𝑊) = ((𝑃 ∨ 𝑆) ∧ 𝑊)
39		eqid 2737	. . . . . 6 ⊢ ((𝑄 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆)) = ((𝑄 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆))
40		eqid 2737	. . . . . 6 ⊢ ((𝑅 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆)) = ((𝑅 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆))
41	36, 6, 7, 8, 9, 10, 37, 38, 39, 40	4atexlemex4 40075	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑄 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆)) = 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
42	36, 6, 7, 8, 9, 10, 37, 38, 39	4atexlemex2 40073	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑄 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆)) ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
43	41, 42	pm2.61dane 3029	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
44	23, 24, 25, 26, 32, 33, 34, 35, 43	syl332anc 1403	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) ∧ 𝑡 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
45	44	rexlimdv3a 3159	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (∃𝑡 ∈ 𝐴 (((𝑃 ∨ 𝑄) ∧ 𝑊) ∨ 𝑡) = (((𝑃 ∨ 𝑆) ∧ 𝑊) ∨ 𝑡) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
46	22, 45	mpd 15	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  HLchlt 39351  LHypclh 39986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lhyp 39990

This theorem is referenced by:  4atexlem7  40077