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Theorem lplnexllnN 39566
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 1193 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝑋𝑃)
2 simpl1 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝐾 ∈ HL)
3 eqid 2737 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
53, 4lplnbase 39536 . . . . 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 39535 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
122, 6, 11syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
131, 12mpbid 232 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)))
14 simpll1 1213 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
15 simpr2l 1233 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
16 simpll3 1215 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
17 simpr1 1195 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑧)
187, 8, 9, 10llnexatN 39523 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧𝑁𝑄𝐴) ∧ 𝑄 𝑧) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
1914, 15, 16, 17, 18syl31anc 1375 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
20 simp1l1 1267 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ HL)
21 simp22r 1294 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝐴)
22 simp3l 1202 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠𝐴)
23 simp1l3 1269 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝐴)
24 simp23l 1295 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 𝑧)
25 simp3rr 1248 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑧 = (𝑄 𝑠))
2625breq2d 5155 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑧𝑟 (𝑄 𝑠)))
2724, 26mtbid 324 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 (𝑄 𝑠))
287, 8, 9atnlej2 39382 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑠𝐴) ∧ ¬ 𝑟 (𝑄 𝑠)) → 𝑟𝑠)
2920, 21, 23, 22, 27, 28syl131anc 1385 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝑠)
308, 9, 10llni2 39514 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) ∧ 𝑟𝑠) → (𝑟 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1375 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑠) ∈ 𝑁)
32 simp3rl 1247 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝑠)
337, 8, 9hlatcon2 39454 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑠𝐴𝑟𝐴) ∧ (𝑄𝑠 ∧ ¬ 𝑟 (𝑄 𝑠))) → ¬ 𝑄 (𝑟 𝑠))
3420, 23, 22, 21, 32, 27, 33syl132anc 1390 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑄 (𝑟 𝑠))
35 simp23r 1296 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = (𝑧 𝑟))
3625oveq1d 7446 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑧 𝑟) = ((𝑄 𝑠) 𝑟))
3720hllatd 39365 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ Lat)
383, 9atbase 39290 . . . . . . . . . . . . 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 39290 . . . . . . . . . . . . 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 39290 . . . . . . . . . . . . 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 18532 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4537, 39, 41, 43, 44syl13anc 1374 . . . . . . . . . . 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 2781 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = ((𝑟 𝑠) 𝑄))
47 breq2 5147 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑄 𝑦𝑄 (𝑟 𝑠)))
4847notbid 318 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 (𝑟 𝑠)))
49 oveq1 7438 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑦 𝑄) = ((𝑟 𝑠) 𝑄))
5049eqeq2d 2748 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = ((𝑟 𝑠) 𝑄)))
5148, 50anbi12d 632 . . . . . . . . . . 11 (𝑦 = (𝑟 𝑠) → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))))
5251rspcev 3622 . . . . . . . . . 10 (((𝑟 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
5331, 34, 46, 52syl12anc 837 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
54533expia 1122 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
5554expd 415 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑠𝐴 → ((𝑄𝑠𝑧 = (𝑄 𝑠)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
5655rexlimdv 3153 . . . . . 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 1355 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
59 simpr2l 1233 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
60 simpr1 1195 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ¬ 𝑄 𝑧)
61 simpll1 1213 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
6261hllatd 39365 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ Lat)
633, 10llnbase 39511 . . . . . . . . . . . 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 1234 . . . . . . . . . . . 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 18493 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 (𝑧 𝑟))
6862, 64, 66, 67syl3anc 1373 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 (𝑧 𝑟))
69 simpr3r 1236 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑟))
7068, 69breqtrrd 5171 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 𝑋)
71 simplr 769 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑋)
72 simpll3 1215 . . . . . . . . . . 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 1214 . . . . . . . . . . 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 18495 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7762, 64, 73, 75, 76syl13anc 1374 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7870, 71, 77mpbi2and 712 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) 𝑋)
793, 8latjcl 18484 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 𝑄) ∈ (Base‘𝐾))
8062, 64, 73, 79syl3anc 1373 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ (Base‘𝐾))
81 eqid 2737 . . . . . . . . . . . . 13 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
823, 7, 8, 81, 9cvr1 39412 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄𝐴) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8361, 64, 72, 82syl3anc 1373 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8460, 83mpbid 232 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧( ⋖ ‘𝐾)(𝑧 𝑄))
853, 81, 10, 4lplni 39534 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ (Base‘𝐾) ∧ 𝑧𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 𝑄)) → (𝑧 𝑄) ∈ 𝑃)
8661, 80, 59, 84, 85syl31anc 1375 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ 𝑃)
877, 4lplncmp 39564 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ 𝑃𝑋𝑃) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8861, 86, 74, 87syl3anc 1373 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8978, 88mpbid 232 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) = 𝑋)
9089eqcomd 2743 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑄))
91 breq2 5147 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑄 𝑦𝑄 𝑧))
9291notbid 318 . . . . . . . 8 (𝑦 = 𝑧 → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 𝑧))
93 oveq1 7438 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 𝑄) = (𝑧 𝑄))
9493eqeq2d 2748 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = (𝑧 𝑄)))
9592, 94anbi12d 632 . . . . . . 7 (𝑦 = 𝑧 → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))))
9695rspcev 3622 . . . . . 6 ((𝑧𝑁 ∧ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
9759, 60, 90, 96syl12anc 837 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
98973exp2 1355 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (¬ 𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
9958, 98pm2.61d 179 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
10099rexlimdvv 3212 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  Latclat 18476  ccvr 39263  Atomscatm 39264  HLchlt 39351  LLinesclln 39493  LPlanesclpl 39494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501
This theorem is referenced by: (None)
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