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Theorem 2llnjN 36821
 Description: The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2llnj.l = (le‘𝐾)
2llnj.j = (join‘𝐾)
2llnj.n 𝑁 = (LLines‘𝐾)
2llnj.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
2llnjN ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) = 𝑊)

Proof of Theorem 2llnjN
Dummy variables 𝑟 𝑞 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2llnj.j . . . . . . . 8 = (join‘𝐾)
3 eqid 2822 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
4 2llnj.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
51, 2, 3, 4islln2 36765 . . . . . . 7 (𝐾 ∈ HL → (𝑋𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)))))
6 simpr 488 . . . . . . 7 ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)))
75, 6syl6bi 256 . . . . . 6 (𝐾 ∈ HL → (𝑋𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))))
81, 2, 3, 4islln2 36765 . . . . . . 7 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
9 simpr 488 . . . . . . 7 ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))) → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))
108, 9syl6bi 256 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
117, 10anim12d 611 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑁𝑌𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
1211imp 410 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
13123adantr3 1168 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
14133adant3 1129 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
15 simp2rr 1240 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑋 = (𝑞 𝑟))
16 simp3rr 1244 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑌 = (𝑠 𝑡))
1715, 16oveq12d 7158 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → (𝑋 𝑌) = ((𝑞 𝑟) (𝑠 𝑡)))
18 simp13 1202 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → (𝑋 𝑊𝑌 𝑊𝑋𝑌))
19 breq1 5045 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋 𝑊 ↔ (𝑞 𝑟) 𝑊))
20 neeq1 3073 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋𝑌 ↔ (𝑞 𝑟) ≠ 𝑌))
2119, 203anbi13d 1435 . . . . . . . . . . . . . 14 (𝑋 = (𝑞 𝑟) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌)))
22 breq1 5045 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → (𝑌 𝑊 ↔ (𝑠 𝑡) 𝑊))
23 neeq2 3074 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → ((𝑞 𝑟) ≠ 𝑌 ↔ (𝑞 𝑟) ≠ (𝑠 𝑡)))
2422, 233anbi23d 1436 . . . . . . . . . . . . . 14 (𝑌 = (𝑠 𝑡) → (((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2521, 24sylan9bb 513 . . . . . . . . . . . . 13 ((𝑋 = (𝑞 𝑟) ∧ 𝑌 = (𝑠 𝑡)) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2615, 16, 25syl2anc 587 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2718, 26mpbid 235 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡)))
28 simp11 1200 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝐾 ∈ HL)
29 simp123 1304 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑊𝑃)
30 simp2ll 1237 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑞 ∈ (Atoms‘𝐾))
31 simp2lr 1238 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑟 ∈ (Atoms‘𝐾))
32 simp2rl 1239 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑞𝑟)
33 simp3ll 1241 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑠 ∈ (Atoms‘𝐾))
34 simp3lr 1242 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑡 ∈ (Atoms‘𝐾))
35 simp3rl 1243 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → 𝑠𝑡)
36 2llnj.l . . . . . . . . . . . . . 14 = (le‘𝐾)
37 2llnj.p . . . . . . . . . . . . . 14 𝑃 = (LPlanes‘𝐾)
3836, 2, 3, 4, 372llnjaN 36820 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝑃) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑞𝑟) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑠𝑡)) ∧ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))) → ((𝑞 𝑟) (𝑠 𝑡)) = 𝑊)
3938ex 416 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝑃) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾) ∧ 𝑞𝑟) ∧ (𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾) ∧ 𝑠𝑡)) → (((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡)) → ((𝑞 𝑟) (𝑠 𝑡)) = 𝑊))
4028, 29, 30, 31, 32, 33, 34, 35, 39syl233anc 1396 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → (((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡)) → ((𝑞 𝑟) (𝑠 𝑡)) = 𝑊))
4127, 40mpd 15 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → ((𝑞 𝑟) (𝑠 𝑡)) = 𝑊)
4217, 41eqtrd 2857 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) ∧ ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡)))) → (𝑋 𝑌) = 𝑊)
43423exp 1116 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) → (((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡))) → (𝑋 𝑌) = 𝑊)))
44433impib 1113 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) → (((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) ∧ (𝑠𝑡𝑌 = (𝑠 𝑡))) → (𝑋 𝑌) = 𝑊))
4544expd 419 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) → ((𝑠 ∈ (Atoms‘𝐾) ∧ 𝑡 ∈ (Atoms‘𝐾)) → ((𝑠𝑡𝑌 = (𝑠 𝑡)) → (𝑋 𝑌) = 𝑊)))
4645rexlimdvv 3279 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞𝑟𝑋 = (𝑞 𝑟))) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)) → (𝑋 𝑌) = 𝑊))
47463exp 1116 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑞𝑟𝑋 = (𝑞 𝑟)) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)) → (𝑋 𝑌) = 𝑊))))
4847rexlimdvv 3279 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) → (∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)) → (𝑋 𝑌) = 𝑊)))
4948impd 414 . 2 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → ((∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))) → (𝑋 𝑌) = 𝑊))
5014, 49mpd 15 1 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃) ∧ (𝑋 𝑊𝑌 𝑊𝑋𝑌)) → (𝑋 𝑌) = 𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∃wrex 3131   class class class wbr 5042  ‘cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  Atomscatm 36517  HLchlt 36604  LLinesclln 36745  LPlanesclpl 36746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-lat 17647  df-clat 17709  df-oposet 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-llines 36752  df-lplanes 36753 This theorem is referenced by:  2llnm2N  36822
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