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Theorem 4atex 34179
Description: Whenever there are at least 4 atoms under 𝑃 𝑄 (specifically, 𝑃, 𝑄, 𝑟, and (𝑃 𝑄) 𝑊), there are also at least 4 atoms under 𝑃 𝑆. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 33447, our (𝑃 𝑟) = (𝑄 𝑟) is a shorter way to express 𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄). (Contributed by NM, 27-May-2013.)
Hypotheses
Ref Expression
4that.l = (le‘𝐾)
4that.j = (join‘𝐾)
4that.a 𝐴 = (Atoms‘𝐾)
4that.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
4atex (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Distinct variable groups:   𝑧,𝑟,𝐴   𝐻,𝑟   ,𝑟,𝑧   𝐾,𝑟,𝑧   ,𝑟,𝑧   𝑃,𝑟,𝑧   𝑄,𝑟,𝑧   𝑆,𝑟,𝑧   𝑊,𝑟,𝑧
Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem 4atex
StepHypRef Expression
1 simp21l 1170 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
21ad2antrr 757 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆 = 𝑃) → 𝑃𝐴)
3 simp21r 1171 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑃 𝑊)
43ad2antrr 757 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆 = 𝑃) → ¬ 𝑃 𝑊)
5 oveq1 6530 . . . . . 6 (𝑃 = 𝑆 → (𝑃 𝑃) = (𝑆 𝑃))
65eqcoms 2613 . . . . 5 (𝑆 = 𝑃 → (𝑃 𝑃) = (𝑆 𝑃))
76adantl 480 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆 = 𝑃) → (𝑃 𝑃) = (𝑆 𝑃))
8 breq1 4576 . . . . . . 7 (𝑧 = 𝑃 → (𝑧 𝑊𝑃 𝑊))
98notbid 306 . . . . . 6 (𝑧 = 𝑃 → (¬ 𝑧 𝑊 ↔ ¬ 𝑃 𝑊))
10 oveq2 6531 . . . . . . 7 (𝑧 = 𝑃 → (𝑃 𝑧) = (𝑃 𝑃))
11 oveq2 6531 . . . . . . 7 (𝑧 = 𝑃 → (𝑆 𝑧) = (𝑆 𝑃))
1210, 11eqeq12d 2620 . . . . . 6 (𝑧 = 𝑃 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝑃) = (𝑆 𝑃)))
139, 12anbi12d 742 . . . . 5 (𝑧 = 𝑃 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝑃 𝑊 ∧ (𝑃 𝑃) = (𝑆 𝑃))))
1413rspcev 3277 . . . 4 ((𝑃𝐴 ∧ (¬ 𝑃 𝑊 ∧ (𝑃 𝑃) = (𝑆 𝑃))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
152, 4, 7, 14syl12anc 1315 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆 = 𝑃) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
16 simpl3r 1109 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
1716ad2antrr 757 . . . . 5 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆 = 𝑄) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
18 oveq1 6530 . . . . . . . . . 10 (𝑆 = 𝑄 → (𝑆 𝑧) = (𝑄 𝑧))
1918eqeq2d 2615 . . . . . . . . 9 (𝑆 = 𝑄 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝑧) = (𝑄 𝑧)))
2019anbi2d 735 . . . . . . . 8 (𝑆 = 𝑄 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧))))
2120rexbidv 3029 . . . . . . 7 (𝑆 = 𝑄 → (∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧))))
22 breq1 4576 . . . . . . . . . 10 (𝑟 = 𝑧 → (𝑟 𝑊𝑧 𝑊))
2322notbid 306 . . . . . . . . 9 (𝑟 = 𝑧 → (¬ 𝑟 𝑊 ↔ ¬ 𝑧 𝑊))
24 oveq2 6531 . . . . . . . . . 10 (𝑟 = 𝑧 → (𝑃 𝑟) = (𝑃 𝑧))
25 oveq2 6531 . . . . . . . . . 10 (𝑟 = 𝑧 → (𝑄 𝑟) = (𝑄 𝑧))
2624, 25eqeq12d 2620 . . . . . . . . 9 (𝑟 = 𝑧 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑧) = (𝑄 𝑧)))
2723, 26anbi12d 742 . . . . . . . 8 (𝑟 = 𝑧 → ((¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ (¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧))))
2827cbvrexv 3143 . . . . . . 7 (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧)))
2921, 28syl6rbbr 277 . . . . . 6 (𝑆 = 𝑄 → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))
3029adantl 480 . . . . 5 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆 = 𝑄) → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))
3117, 30mpbid 220 . . . 4 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆 = 𝑄) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
32 simp22l 1172 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
3332ad3antrrr 761 . . . . 5 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑄𝐴)
34 simp22r 1173 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑄 𝑊)
3534ad3antrrr 761 . . . . 5 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → ¬ 𝑄 𝑊)
36 simp3l 1081 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
3736necomd 2832 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝑃)
3837ad3antrrr 761 . . . . . 6 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑄𝑃)
39 simpr 475 . . . . . . 7 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑆𝑄)
4039necomd 2832 . . . . . 6 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑄𝑆)
41 simpllr 794 . . . . . . 7 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑆 (𝑃 𝑄))
42 simp1l 1077 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
43 hlcvl 33463 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
4442, 43syl 17 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ CvLat)
4544ad3antrrr 761 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝐾 ∈ CvLat)
46 simp23 1088 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆𝐴)
4746ad3antrrr 761 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑆𝐴)
481ad3antrrr 761 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑃𝐴)
49 simplr 787 . . . . . . . 8 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑆𝑃)
50 4that.l . . . . . . . . 9 = (le‘𝐾)
51 4that.j . . . . . . . . 9 = (join‘𝐾)
52 4that.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
5350, 51, 52cvlatexch1 33440 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑆𝐴𝑄𝐴𝑃𝐴) ∧ 𝑆𝑃) → (𝑆 (𝑃 𝑄) → 𝑄 (𝑃 𝑆)))
5445, 47, 33, 48, 49, 53syl131anc 1330 . . . . . . 7 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → (𝑆 (𝑃 𝑄) → 𝑄 (𝑃 𝑆)))
5541, 54mpd 15 . . . . . 6 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑄 (𝑃 𝑆))
5649necomd 2832 . . . . . . 7 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → 𝑃𝑆)
5752, 50, 51cvlsupr2 33447 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝑄𝐴) ∧ 𝑃𝑆) → ((𝑃 𝑄) = (𝑆 𝑄) ↔ (𝑄𝑃𝑄𝑆𝑄 (𝑃 𝑆))))
5845, 48, 47, 33, 56, 57syl131anc 1330 . . . . . 6 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → ((𝑃 𝑄) = (𝑆 𝑄) ↔ (𝑄𝑃𝑄𝑆𝑄 (𝑃 𝑆))))
5938, 40, 55, 58mpbir3and 1237 . . . . 5 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → (𝑃 𝑄) = (𝑆 𝑄))
60 breq1 4576 . . . . . . . 8 (𝑧 = 𝑄 → (𝑧 𝑊𝑄 𝑊))
6160notbid 306 . . . . . . 7 (𝑧 = 𝑄 → (¬ 𝑧 𝑊 ↔ ¬ 𝑄 𝑊))
62 oveq2 6531 . . . . . . . 8 (𝑧 = 𝑄 → (𝑃 𝑧) = (𝑃 𝑄))
63 oveq2 6531 . . . . . . . 8 (𝑧 = 𝑄 → (𝑆 𝑧) = (𝑆 𝑄))
6462, 63eqeq12d 2620 . . . . . . 7 (𝑧 = 𝑄 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝑄) = (𝑆 𝑄)))
6561, 64anbi12d 742 . . . . . 6 (𝑧 = 𝑄 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑆 𝑄))))
6665rspcev 3277 . . . . 5 ((𝑄𝐴 ∧ (¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑆 𝑄))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6733, 35, 59, 66syl12anc 1315 . . . 4 ((((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) ∧ 𝑆𝑄) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6831, 67pm2.61dane 2864 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) ∧ 𝑆𝑃) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6915, 68pm2.61dane 2864 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ 𝑆 (𝑃 𝑄)) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
70 simpl1 1056 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
71 simpl2 1057 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴))
72 simpl3l 1108 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑄)
73 simpr 475 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ 𝑆 (𝑃 𝑄))
74 simpl3r 1109 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))
75 4that.h . . . 4 𝐻 = (LHyp‘𝐾)
7650, 51, 52, 754atexlem7 34178 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
7770, 71, 72, 73, 74, 76syl113anc 1329 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
7869, 77pm2.61dan 827 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wrex 2892   class class class wbr 4573  cfv 5786  (class class class)co 6523  lecple 15717  joincjn 16709  Atomscatm 33367  CvLatclc 33369  HLchlt 33454  LHypclh 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-p1 16805  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-llines 33601  df-lplanes 33602  df-lhyp 34091
This theorem is referenced by:  4atex2  34180
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