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Theorem lplnexllnN 35638
Description: Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnexat.l = (le‘𝐾)
lplnexat.j = (join‘𝐾)
lplnexat.a 𝐴 = (Atoms‘𝐾)
lplnexat.n 𝑁 = (LLines‘𝐾)
lplnexat.p 𝑃 = (LPlanes‘𝐾)
Assertion
Ref Expression
lplnexllnN (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
Distinct variable groups:   𝑦,   𝑦,   𝑦,𝑁   𝑦,𝑄   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑦)   𝐾(𝑦)

Proof of Theorem lplnexllnN
Dummy variables 𝑠 𝑟 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1248 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝑋𝑃)
2 simpl1 1246 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝐾 ∈ HL)
3 eqid 2825 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
53, 4lplnbase 35608 . . . . 5 (𝑋𝑃𝑋 ∈ (Base‘𝐾))
61, 5syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝑋 ∈ (Base‘𝐾))
7 lplnexat.l . . . . 5 = (le‘𝐾)
8 lplnexat.j . . . . 5 = (join‘𝐾)
9 lplnexat.a . . . . 5 𝐴 = (Atoms‘𝐾)
10 lplnexat.n . . . . 5 𝑁 = (LLines‘𝐾)
113, 7, 8, 9, 10, 4islpln3 35607 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
122, 6, 11syl2anc 579 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
131, 12mpbid 224 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)))
14 simpll1 1273 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
15 simpr2l 1313 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
16 simpll3 1277 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
17 simpr1 1252 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑧)
187, 8, 9, 10llnexatN 35595 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧𝑁𝑄𝐴) ∧ 𝑄 𝑧) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
1914, 15, 16, 17, 18syl31anc 1496 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
20 simp1l1 1369 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ HL)
21 simp22r 1396 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝐴)
22 simp3l 1262 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠𝐴)
23 simp1l3 1371 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝐴)
24 simp23l 1397 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 𝑧)
25 simp3rr 1332 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑧 = (𝑄 𝑠))
2625breq2d 4887 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑧𝑟 (𝑄 𝑠)))
2724, 26mtbid 316 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 (𝑄 𝑠))
287, 8, 9atnlej2 35454 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑠𝐴) ∧ ¬ 𝑟 (𝑄 𝑠)) → 𝑟𝑠)
2920, 21, 23, 22, 27, 28syl131anc 1506 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝑠)
308, 9, 10llni2 35586 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) ∧ 𝑟𝑠) → (𝑟 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1496 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑠) ∈ 𝑁)
32 simp3rl 1331 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝑠)
337, 8, 9hlatcon2 35526 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑠𝐴𝑟𝐴) ∧ (𝑄𝑠 ∧ ¬ 𝑟 (𝑄 𝑠))) → ¬ 𝑄 (𝑟 𝑠))
3420, 23, 22, 21, 32, 27, 33syl132anc 1511 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑄 (𝑟 𝑠))
35 simp23r 1398 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = (𝑧 𝑟))
3625oveq1d 6925 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑧 𝑟) = ((𝑄 𝑠) 𝑟))
3720hllatd 35438 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ Lat)
383, 9atbase 35363 . . . . . . . . . . . . 13 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
3923, 38syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄 ∈ (Base‘𝐾))
403, 9atbase 35363 . . . . . . . . . . . . 13 (𝑠𝐴𝑠 ∈ (Base‘𝐾))
4122, 40syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠 ∈ (Base‘𝐾))
423, 9atbase 35363 . . . . . . . . . . . . 13 (𝑟𝐴𝑟 ∈ (Base‘𝐾))
4321, 42syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟 ∈ (Base‘𝐾))
443, 8latj31 17459 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4537, 39, 41, 43, 44syl13anc 1495 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4635, 36, 453eqtrd 2865 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = ((𝑟 𝑠) 𝑄))
47 breq2 4879 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑄 𝑦𝑄 (𝑟 𝑠)))
4847notbid 310 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 (𝑟 𝑠)))
49 oveq1 6917 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑦 𝑄) = ((𝑟 𝑠) 𝑄))
5049eqeq2d 2835 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = ((𝑟 𝑠) 𝑄)))
5148, 50anbi12d 624 . . . . . . . . . . 11 (𝑦 = (𝑟 𝑠) → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))))
5251rspcev 3526 . . . . . . . . . 10 (((𝑟 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
5331, 34, 46, 52syl12anc 870 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
54533expia 1154 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
5554expd 406 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑠𝐴 → ((𝑄𝑠𝑧 = (𝑄 𝑠)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
5655rexlimdv 3239 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
5719, 56mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
58573exp2 1467 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
59 simpr2l 1313 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
60 simpr1 1252 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ¬ 𝑄 𝑧)
61 simpll1 1273 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
6261hllatd 35438 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ Lat)
633, 10llnbase 35583 . . . . . . . . . . . 12 (𝑧𝑁𝑧 ∈ (Base‘𝐾))
6459, 63syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 ∈ (Base‘𝐾))
65 simpr2r 1315 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑟𝐴)
6665, 42syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑟 ∈ (Base‘𝐾))
673, 7, 8latlej1 17420 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 (𝑧 𝑟))
6862, 64, 66, 67syl3anc 1494 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 (𝑧 𝑟))
69 simpr3r 1319 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑟))
7068, 69breqtrrd 4903 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 𝑋)
71 simplr 785 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑋)
72 simpll3 1277 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
7372, 38syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 ∈ (Base‘𝐾))
74 simpll2 1275 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋𝑃)
7574, 5syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 ∈ (Base‘𝐾))
763, 7, 8latjle12 17422 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7762, 64, 73, 75, 76syl13anc 1495 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7870, 71, 77mpbi2and 703 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) 𝑋)
793, 8latjcl 17411 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 𝑄) ∈ (Base‘𝐾))
8062, 64, 73, 79syl3anc 1494 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ (Base‘𝐾))
81 eqid 2825 . . . . . . . . . . . . 13 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
823, 7, 8, 81, 9cvr1 35484 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄𝐴) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8361, 64, 72, 82syl3anc 1494 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8460, 83mpbid 224 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧( ⋖ ‘𝐾)(𝑧 𝑄))
853, 81, 10, 4lplni 35606 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ (Base‘𝐾) ∧ 𝑧𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 𝑄)) → (𝑧 𝑄) ∈ 𝑃)
8661, 80, 59, 84, 85syl31anc 1496 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ 𝑃)
877, 4lplncmp 35636 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ 𝑃𝑋𝑃) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8861, 86, 74, 87syl3anc 1494 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8978, 88mpbid 224 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) = 𝑋)
9089eqcomd 2831 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑄))
91 breq2 4879 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑄 𝑦𝑄 𝑧))
9291notbid 310 . . . . . . . 8 (𝑦 = 𝑧 → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 𝑧))
93 oveq1 6917 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 𝑄) = (𝑧 𝑄))
9493eqeq2d 2835 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = (𝑧 𝑄)))
9592, 94anbi12d 624 . . . . . . 7 (𝑦 = 𝑧 → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))))
9695rspcev 3526 . . . . . 6 ((𝑧𝑁 ∧ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
9759, 60, 90, 96syl12anc 870 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
98973exp2 1467 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (¬ 𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
9958, 98pm2.61d 172 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
10099rexlimdvv 3247 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
10113, 100mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wrex 3118   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  Latclat 17405  ccvr 35336  Atomscatm 35337  HLchlt 35424  LLinesclln 35565  LPlanesclpl 35566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-llines 35572  df-lplanes 35573
This theorem is referenced by: (None)
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