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Theorem 2llnjN 35356
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 2760 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2llnj.j . . . . . . . 8 = (join‘𝐾)
3 eqid 2760 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
4 2llnj.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
51, 2, 3, 4islln2 35300 . . . . . . 7 (𝐾 ∈ HL → (𝑋𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)))))
6 simpr 479 . . . . . . 7 ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)))
75, 6syl6bi 243 . . . . . 6 (𝐾 ∈ HL → (𝑋𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))))
81, 2, 3, 4islln2 35300 . . . . . . 7 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
9 simpr 479 . . . . . . 7 ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))) → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))
108, 9syl6bi 243 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
117, 10anim12d 587 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑁𝑌𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
1211imp 444 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
13123adantr3 1177 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
14133adant3 1127 . 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 1310 . . . . . . . . . . 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 1314 . . . . . . . . . . 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 6831 . . . . . . . . . 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 1248 . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋 𝑊 ↔ (𝑞 𝑟) 𝑊))
20 neeq1 2994 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋𝑌 ↔ (𝑞 𝑟) ≠ 𝑌))
2119, 203anbi13d 1550 . . . . . . . . . . . . . 14 (𝑋 = (𝑞 𝑟) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌)))
22 breq1 4807 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → (𝑌 𝑊 ↔ (𝑠 𝑡) 𝑊))
23 neeq2 2995 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → ((𝑞 𝑟) ≠ 𝑌 ↔ (𝑞 𝑟) ≠ (𝑠 𝑡)))
2422, 233anbi23d 1551 . . . . . . . . . . . . . 14 (𝑌 = (𝑠 𝑡) → (((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2521, 24sylan9bb 738 . . . . . . . . . . . . 13 ((𝑋 = (𝑞 𝑟) ∧ 𝑌 = (𝑠 𝑡)) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2615, 16, 25syl2anc 696 . . . . . . . . . . . 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 222 . . . . . . . . . . 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 1246 . . . . . . . . . . . 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 1392 . . . . . . . . . . . 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 1307 . . . . . . . . . . . 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 1308 . . . . . . . . . . . 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 1309 . . . . . . . . . . . 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 1311 . . . . . . . . . . . 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 1312 . . . . . . . . . . . 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 1313 . . . . . . . . . . . 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 35355 . . . . . . . . . . . . 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 449 . . . . . . . . . . . 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 1506 . . . . . . . . . . 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 2794 . . . . . . . . 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 1113 . . . . . . . 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 1109 . . . . . . 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 451 . . . . . 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 3175 . . . . 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 1113 . . . 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 3175 . . 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 446 . 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  Atomscatm 35053  HLchlt 35140  LLinesclln 35280  LPlanesclpl 35281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288
This theorem is referenced by:  2llnm2N  35357
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