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Theorem 4atexlemex6 36094
Description: Lemma for 4atexlem7 36095. (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 ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ∃𝑧𝐴𝑧 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.
StepHypRef Expression
1 simp11l 1384 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ HL)
2 simp11 1261 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simp12 1262 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
4 simp13l 1388 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑄𝐴)
5 simp32 1268 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝑄)
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)
116, 7, 8, 9, 10lhpat 36063 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
122, 3, 4, 5, 11syl112anc 1494 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 𝑄) 𝑊) ∈ 𝐴)
13 simp2r 1258 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝐴)
14 simp12l 1386 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝐴)
15 simp33 1269 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
166, 7, 9atnlej1 35399 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝑃)
171, 13, 14, 4, 15, 16syl131anc 1503 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝑃)
1817necomd 3027 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝑆)
196, 7, 8, 9, 10lhpat 36063 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴𝑃𝑆)) → ((𝑃 𝑆) 𝑊) ∈ 𝐴)
202, 3, 13, 18, 19syl112anc 1494 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 𝑆) 𝑊) ∈ 𝐴)
217, 9hlsupr2 35407 . . 3 ((𝐾 ∈ HL ∧ ((𝑃 𝑄) 𝑊) ∈ 𝐴 ∧ ((𝑃 𝑆) 𝑊) ∈ 𝐴) → ∃𝑡𝐴 (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))
221, 12, 20, 21syl3anc 1491 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ∃𝑡𝐴 (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))
23 simp111 1402 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
24 simp112 1403 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
25 simp113 1404 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
26 simp12r 1387 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → 𝑆𝐴)
27 simp2ll 1322 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅𝐴)
28273ad2ant1 1164 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → 𝑅𝐴)
29 simp2lr 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑅 𝑊)
30293ad2ant1 1164 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → ¬ 𝑅 𝑊)
31 simp131 1408 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → (𝑃 𝑅) = (𝑄 𝑅))
3228, 30, 313jca 1159 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)))
33 3simpc 1183 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → (𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)))
34 simp132 1409 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → 𝑃𝑄)
35 simp133 1410 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → ¬ 𝑆 (𝑃 𝑄))
36 biid 253 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ↔ (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))))
37 eqid 2800 . . . . . 6 ((𝑃 𝑄) 𝑊) = ((𝑃 𝑄) 𝑊)
38 eqid 2800 . . . . . 6 ((𝑃 𝑆) 𝑊) = ((𝑃 𝑆) 𝑊)
39 eqid 2800 . . . . . 6 ((𝑄 𝑡) (𝑃 𝑆)) = ((𝑄 𝑡) (𝑃 𝑆))
40 eqid 2800 . . . . . 6 ((𝑅 𝑡) (𝑃 𝑆)) = ((𝑅 𝑡) (𝑃 𝑆))
4136, 6, 7, 8, 9, 10, 37, 38, 39, 404atexlemex4 36093 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ ((𝑄 𝑡) (𝑃 𝑆)) = 𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
4236, 6, 7, 8, 9, 10, 37, 38, 394atexlemex2 36091 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ ((𝑄 𝑡) (𝑃 𝑆)) ≠ 𝑆) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
4341, 42pm2.61dane 3059 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ (𝑃 𝑅) = (𝑄 𝑅)) ∧ (𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡))) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
4423, 24, 25, 26, 32, 33, 34, 35, 43syl332anc 1521 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝑡𝐴 ∧ (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡)) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
4544rexlimdv3a 3215 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → (∃𝑡𝐴 (((𝑃 𝑄) 𝑊) 𝑡) = (((𝑃 𝑆) 𝑊) 𝑡) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))
4622, 45mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑆𝐴) ∧ ((𝑃 𝑅) = (𝑄 𝑅) ∧ 𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2972  wrex 3091   class class class wbr 4844  cfv 6102  (class class class)co 6879  lecple 16273  joincjn 17258  meetcmee 17259  Atomscatm 35283  HLchlt 35370  LHypclh 36004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-proset 17242  df-poset 17260  df-plt 17272  df-lub 17288  df-glb 17289  df-join 17290  df-meet 17291  df-p0 17353  df-p1 17354  df-lat 17360  df-clat 17422  df-oposet 35196  df-ol 35198  df-oml 35199  df-covers 35286  df-ats 35287  df-atl 35318  df-cvlat 35342  df-hlat 35371  df-llines 35518  df-lplanes 35519  df-lhyp 36008
This theorem is referenced by:  4atexlem7  36095
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