Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atex Structured version   Visualization version   GIF version

Theorem 4atex 35856
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 35123, 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 1382 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝐴)
21ad2antrr 708 . . . 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 1383 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑃 𝑊)
43ad2antrr 708 . . . 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 6881 . . . . . 6 (𝑃 = 𝑆 → (𝑃 𝑃) = (𝑆 𝑃))
65eqcoms 2814 . . . . 5 (𝑆 = 𝑃 → (𝑃 𝑃) = (𝑆 𝑃))
76adantl 469 . . . 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 4847 . . . . . . 7 (𝑧 = 𝑃 → (𝑧 𝑊𝑃 𝑊))
98notbid 309 . . . . . 6 (𝑧 = 𝑃 → (¬ 𝑧 𝑊 ↔ ¬ 𝑃 𝑊))
10 oveq2 6882 . . . . . . 7 (𝑧 = 𝑃 → (𝑃 𝑧) = (𝑃 𝑃))
11 oveq2 6882 . . . . . . 7 (𝑧 = 𝑃 → (𝑆 𝑧) = (𝑆 𝑃))
1210, 11eqeq12d 2821 . . . . . 6 (𝑧 = 𝑃 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝑃) = (𝑆 𝑃)))
139, 12anbi12d 618 . . . . 5 (𝑧 = 𝑃 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝑃 𝑊 ∧ (𝑃 𝑃) = (𝑆 𝑃))))
1413rspcev 3502 . . . 4 ((𝑃𝐴 ∧ (¬ 𝑃 𝑊 ∧ (𝑃 𝑃) = (𝑆 𝑃))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
152, 4, 7, 14syl12anc 856 . . 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 1296 . . . . . 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 708 . . . . 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 6881 . . . . . . . . . 10 (𝑆 = 𝑄 → (𝑆 𝑧) = (𝑄 𝑧))
1918eqeq2d 2816 . . . . . . . . 9 (𝑆 = 𝑄 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝑧) = (𝑄 𝑧)))
2019anbi2d 616 . . . . . . . 8 (𝑆 = 𝑄 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧))))
2120rexbidv 3240 . . . . . . 7 (𝑆 = 𝑄 → (∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧))))
22 breq1 4847 . . . . . . . . . 10 (𝑟 = 𝑧 → (𝑟 𝑊𝑧 𝑊))
2322notbid 309 . . . . . . . . 9 (𝑟 = 𝑧 → (¬ 𝑟 𝑊 ↔ ¬ 𝑧 𝑊))
24 oveq2 6882 . . . . . . . . . 10 (𝑟 = 𝑧 → (𝑃 𝑟) = (𝑃 𝑧))
25 oveq2 6882 . . . . . . . . . 10 (𝑟 = 𝑧 → (𝑄 𝑟) = (𝑄 𝑧))
2624, 25eqeq12d 2821 . . . . . . . . 9 (𝑟 = 𝑧 → ((𝑃 𝑟) = (𝑄 𝑟) ↔ (𝑃 𝑧) = (𝑄 𝑧)))
2723, 26anbi12d 618 . . . . . . . 8 (𝑟 = 𝑧 → ((¬ 𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ (¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧))))
2827cbvrexv 3361 . . . . . . 7 (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑄 𝑧)))
2921, 28syl6rbbr 281 . . . . . 6 (𝑆 = 𝑄 → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ↔ ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧))))
3029adantl 469 . . . . 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 223 . . . 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 1384 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝐴)
3332ad3antrrr 712 . . . . 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 1385 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ¬ 𝑄 𝑊)
3534ad3antrrr 712 . . . . 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 1251 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑃𝑄)
3736necomd 3033 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑄𝑃)
3837ad3antrrr 712 . . . . . 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 473 . . . . . . 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 3033 . . . . . 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 784 . . . . . . 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 1247 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝐾 ∈ HL)
43 hlcvl 35139 . . . . . . . . . 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 712 . . . . . . . 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 1258 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → 𝑆𝐴)
4746ad3antrrr 712 . . . . . . . 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 712 . . . . . . . 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 776 . . . . . . . 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 35116 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑆𝐴𝑄𝐴𝑃𝐴) ∧ 𝑆𝑃) → (𝑆 (𝑃 𝑄) → 𝑄 (𝑃 𝑆)))
5445, 47, 33, 48, 49, 53syl131anc 1495 . . . . . . 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 3033 . . . . . . 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 35123 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝑄𝐴) ∧ 𝑃𝑆) → ((𝑃 𝑄) = (𝑆 𝑄) ↔ (𝑄𝑃𝑄𝑆𝑄 (𝑃 𝑆))))
5845, 48, 47, 33, 56, 57syl131anc 1495 . . . . . 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 1435 . . . . 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 4847 . . . . . . . 8 (𝑧 = 𝑄 → (𝑧 𝑊𝑄 𝑊))
6160notbid 309 . . . . . . 7 (𝑧 = 𝑄 → (¬ 𝑧 𝑊 ↔ ¬ 𝑄 𝑊))
62 oveq2 6882 . . . . . . . 8 (𝑧 = 𝑄 → (𝑃 𝑧) = (𝑃 𝑄))
63 oveq2 6882 . . . . . . . 8 (𝑧 = 𝑄 → (𝑆 𝑧) = (𝑆 𝑄))
6462, 63eqeq12d 2821 . . . . . . 7 (𝑧 = 𝑄 → ((𝑃 𝑧) = (𝑆 𝑧) ↔ (𝑃 𝑄) = (𝑆 𝑄)))
6561, 64anbi12d 618 . . . . . 6 (𝑧 = 𝑄 → ((¬ 𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)) ↔ (¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑆 𝑄))))
6665rspcev 3502 . . . . 5 ((𝑄𝐴 ∧ (¬ 𝑄 𝑊 ∧ (𝑃 𝑄) = (𝑆 𝑄))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑃 𝑧) = (𝑆 𝑧)))
6733, 35, 59, 66syl12anc 856 . . . 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 3065 . . 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 3065 . 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 1235 . . 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 1237 . . 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 1294 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ (𝑃𝑄 ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑄)
73 simpr 473 . . 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 1296 . . 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 35855 . . 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 1494 . 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 838 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wrex 3097   class class class wbr 4844  cfv 6101  (class class class)co 6874  lecple 16160  joincjn 17149  Atomscatm 35043  CvLatclc 35045  HLchlt 35130  LHypclh 35764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-proset 17133  df-poset 17151  df-plt 17163  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-p0 17244  df-p1 17245  df-lat 17251  df-clat 17313  df-oposet 34956  df-ol 34958  df-oml 34959  df-covers 35046  df-ats 35047  df-atl 35078  df-cvlat 35102  df-hlat 35131  df-llines 35278  df-lplanes 35279  df-lhyp 35768
This theorem is referenced by:  4atex2  35857
  Copyright terms: Public domain W3C validator