Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnjN Structured version   Visualization version   GIF version

Theorem 2llnjN 34372
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 2621 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
2 2llnj.j . . . . . . . 8 = (join‘𝐾)
3 eqid 2621 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
4 2llnj.n . . . . . . . 8 𝑁 = (LLines‘𝐾)
51, 2, 3, 4islln2 34316 . . . . . . 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 243 . . . . . 6 (𝐾 ∈ HL → (𝑋𝑁 → ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟))))
81, 2, 3, 4islln2 34316 . . . . . . 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 243 . . . . . 6 (𝐾 ∈ HL → (𝑌𝑁 → ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
117, 10anim12d 585 . . . . 5 (𝐾 ∈ HL → ((𝑋𝑁𝑌𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡)))))
1211imp 445 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
13123adantr3 1220 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑊𝑃)) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞𝑟𝑋 = (𝑞 𝑟)) ∧ ∃𝑠 ∈ (Atoms‘𝐾)∃𝑡 ∈ (Atoms‘𝐾)(𝑠𝑡𝑌 = (𝑠 𝑡))))
14133adant3 1079 . 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 1129 . . . . . . . . . . 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 1133 . . . . . . . . . . 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 6633 . . . . . . . . . 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 1091 . . . . . . . . . . . 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 4626 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋 𝑊 ↔ (𝑞 𝑟) 𝑊))
20 neeq1 2852 . . . . . . . . . . . . . . 15 (𝑋 = (𝑞 𝑟) → (𝑋𝑌 ↔ (𝑞 𝑟) ≠ 𝑌))
2119, 203anbi13d 1398 . . . . . . . . . . . . . 14 (𝑋 = (𝑞 𝑟) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌)))
22 breq1 4626 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → (𝑌 𝑊 ↔ (𝑠 𝑡) 𝑊))
23 neeq2 2853 . . . . . . . . . . . . . . 15 (𝑌 = (𝑠 𝑡) → ((𝑞 𝑟) ≠ 𝑌 ↔ (𝑞 𝑟) ≠ (𝑠 𝑡)))
2422, 233anbi23d 1399 . . . . . . . . . . . . . 14 (𝑌 = (𝑠 𝑡) → (((𝑞 𝑟) 𝑊𝑌 𝑊 ∧ (𝑞 𝑟) ≠ 𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2521, 24sylan9bb 735 . . . . . . . . . . . . 13 ((𝑋 = (𝑞 𝑟) ∧ 𝑌 = (𝑠 𝑡)) → ((𝑋 𝑊𝑌 𝑊𝑋𝑌) ↔ ((𝑞 𝑟) 𝑊 ∧ (𝑠 𝑡) 𝑊 ∧ (𝑞 𝑟) ≠ (𝑠 𝑡))))
2615, 16, 25syl2anc 692 . . . . . . . . . . . 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 1089 . . . . . . . . . . . 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 1193 . . . . . . . . . . . 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 1126 . . . . . . . . . . . 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 1127 . . . . . . . . . . . 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 1128 . . . . . . . . . . . 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 1130 . . . . . . . . . . . 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 1131 . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 34371 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 1352 . . . . . . . . . . 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 2655 . . . . . . . . 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 1261 . . . . . . . 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 1259 . . . . . . 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 452 . . . . . 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 3032 . . . . 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 1261 . . . 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 3032 . . 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 447 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  joincjn 16884  Atomscatm 34069  HLchlt 34156  LLinesclln 34296  LPlanesclpl 34297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-preset 16868  df-poset 16886  df-plt 16898  df-lub 16914  df-glb 16915  df-join 16916  df-meet 16917  df-p0 16979  df-lat 16986  df-clat 17048  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304
This theorem is referenced by:  2llnm2N  34373
  Copyright terms: Public domain W3C validator