Theorem 4atexlemex6 40876

Description: Lemma for 4atexlem7 40877. (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 1302	. . 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 1221	. . . 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 1222	. . . 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 1306	. . . 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 1228	. . . 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 40845	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴)
12	2, 3, 4, 5, 11	syl112anc 1400	. . 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 1218	. . . 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 1304	. . . . . 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 1229	. . . . . 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 40181	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ≠ 𝑃)
17	1, 13, 14, 4, 15, 16	syl131anc 1409	. . . . 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 3012	. . . 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 40845	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐴)
20	2, 3, 13, 18, 19	syl112anc 1400	. . 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 40189	. . 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 1397	. 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 1320	. . . 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 1321	. . . 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 1322	. . . 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 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)) → 𝑆 ∈ 𝐴)
27		simp2ll 1258	. . . . . 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 1150	. . . . 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 1259	. . . . . 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 1150	. . . . 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 1326	. . . . 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 1145	. . . 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 1167	. . . 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 1327	. . . 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 1328	. . . 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 264	. . . . . 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 2762	. . . . . 6 ⊢ ((𝑃 ∨ 𝑄) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)
38		eqid 2762	. . . . . 6 ⊢ ((𝑃 ∨ 𝑆) ∧ 𝑊) = ((𝑃 ∨ 𝑆) ∧ 𝑊)
39		eqid 2762	. . . . . 6 ⊢ ((𝑄 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆)) = ((𝑄 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆))
40		eqid 2762	. . . . . 6 ⊢ ((𝑅 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆)) = ((𝑅 ∨ 𝑡) ∧ (𝑃 ∨ 𝑆))
41	36, 6, 7, 8, 9, 10, 37, 38, 39, 40	4atexlemex4 40875	. . . . 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 40873	. . . . 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 3044	. . . 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 1427	. . 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 3169	. 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 16	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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  lecple 17323  joincjn 18373  meetcmee 18374  Atomscatm 40065  HLchlt 40152  LHypclh 40786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18356  df-poset 18375  df-plt 18390  df-lub 18406  df-glb 18407  df-join 18408  df-meet 18409  df-p0 18485  df-p1 18486  df-lat 18494  df-clat 18561  df-oposet 39978  df-ol 39980  df-oml 39981  df-covers 40068  df-ats 40069  df-atl 40100  df-cvlat 40124  df-hlat 40153  df-llines 40300  df-lplanes 40301  df-lhyp 40790

This theorem is used by:  4atexlem7  40877