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Theorem 2llnjN 36154
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 2778 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2llnj.j . . . . . . . 8 = (join‘𝐾)
3 eqid 2778 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
4 2llnj.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
51, 2, 3, 4islln2 36098 . . . . . . 7 (𝐾 ∈ HL → (𝑋𝑁 ↔ (𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)))))
6 simpr 477 . . . . . . 7 ((𝑋 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))) → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)))
75, 6syl6bi 245 . . . . . 6 (𝐾 ∈ HL → (𝑋𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))))
81, 2, 3, 4islln2 36098 . . . . . . 7 (𝐾 ∈ HL → (𝑌𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
9 simpr 477 . . . . . . 7 ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))) → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))
108, 9syl6bi 245 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
117, 10anim12d 599 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑁𝑌𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
1211imp 398 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
13123adantr3 1151 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
14133adant3 1112 . 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 1223 . . . . . . . . . . 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 1227 . . . . . . . . . . 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 6994 . . . . . . . . . 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 1185 . . . . . . . . . . . 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 4932 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋 𝑊 ↔ (𝑞 𝑟) 𝑊))
20 neeq1 3029 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋𝑌 ↔ (𝑞 𝑟) ≠ 𝑌))
2119, 203anbi13d 1417 . . . . . . . . . . . . . 14 (𝑋 = (𝑞 𝑟) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌)))
22 breq1 4932 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → (𝑌 𝑊 ↔ (𝑠 𝑡) 𝑊))
23 neeq2 3030 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → ((𝑞 𝑟) ≠ 𝑌 ↔ (𝑞 𝑟) ≠ (𝑠 𝑡)))
2422, 233anbi23d 1418 . . . . . . . . . . . . . 14 (𝑌 = (𝑠 𝑡) → (((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2521, 24sylan9bb 502 . . . . . . . . . . . . 13 ((𝑋 = (𝑞 𝑟) ∧ 𝑌 = (𝑠 𝑡)) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2615, 16, 25syl2anc 576 . . . . . . . . . . . 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 224 . . . . . . . . . . 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 1183 . . . . . . . . . . . 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 1287 . . . . . . . . . . . 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 1220 . . . . . . . . . . . 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 1221 . . . . . . . . . . . 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 1222 . . . . . . . . . . . 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 1224 . . . . . . . . . . . 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 1225 . . . . . . . . . . . 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 1226 . . . . . . . . . . . 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 36153 . . . . . . . . . . . . 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 405 . . . . . . . . . . . 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 1379 . . . . . . . . . . 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 2814 . . . . . . . . 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 1099 . . . . . . . 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 1096 . . . . . . 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 408 . . . . . 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 3238 . . . . 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 1099 . . . 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 3238 . . 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 402 . 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 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  wrex 3089   class class class wbr 4929  cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  Atomscatm 35850  HLchlt 35937  LLinesclln 36078  LPlanesclpl 36079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-lat 17514  df-clat 17576  df-oposet 35763  df-ol 35765  df-oml 35766  df-covers 35853  df-ats 35854  df-atl 35885  df-cvlat 35909  df-hlat 35938  df-llines 36085  df-lplanes 36086
This theorem is referenced by:  2llnm2N  36155
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