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Theorem lplnexllnN 35362
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 1237 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝑋𝑃)
2 simpl1 1235 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝐾 ∈ HL)
3 eqid 2817 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
53, 4lplnbase 35332 . . . . 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 35331 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
122, 6, 11syl2anc 575 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
131, 12mpbid 223 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)))
14 simpll1 1262 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
15 simpr2l 1302 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
16 simpll3 1266 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
17 simpr1 1241 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑧)
187, 8, 9, 10llnexatN 35319 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧𝑁𝑄𝐴) ∧ 𝑄 𝑧) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
1914, 15, 16, 17, 18syl31anc 1485 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
20 simp1l1 1358 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ HL)
21 simp22r 1385 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝐴)
22 simp3l 1251 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠𝐴)
23 simp1l3 1360 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝐴)
24 simp23l 1386 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 𝑧)
25 simp3rr 1321 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑧 = (𝑄 𝑠))
2625breq2d 4867 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑧𝑟 (𝑄 𝑠)))
2724, 26mtbid 315 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 (𝑄 𝑠))
287, 8, 9atnlej2 35178 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑠𝐴) ∧ ¬ 𝑟 (𝑄 𝑠)) → 𝑟𝑠)
2920, 21, 23, 22, 27, 28syl131anc 1495 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝑠)
308, 9, 10llni2 35310 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) ∧ 𝑟𝑠) → (𝑟 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1485 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑠) ∈ 𝑁)
32 simp3rl 1320 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝑠)
337, 8, 9hlatcon2 35250 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑠𝐴𝑟𝐴) ∧ (𝑄𝑠 ∧ ¬ 𝑟 (𝑄 𝑠))) → ¬ 𝑄 (𝑟 𝑠))
3420, 23, 22, 21, 32, 27, 33syl132anc 1500 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑄 (𝑟 𝑠))
35 simp23r 1387 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = (𝑧 𝑟))
3625oveq1d 6898 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑧 𝑟) = ((𝑄 𝑠) 𝑟))
3720hllatd 35162 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ Lat)
383, 9atbase 35087 . . . . . . . . . . . . 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 35087 . . . . . . . . . . . . 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 35087 . . . . . . . . . . . . 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 17323 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4537, 39, 41, 43, 44syl13anc 1484 . . . . . . . . . . 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 2855 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = ((𝑟 𝑠) 𝑄))
47 breq2 4859 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑄 𝑦𝑄 (𝑟 𝑠)))
4847notbid 309 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 (𝑟 𝑠)))
49 oveq1 6890 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑦 𝑄) = ((𝑟 𝑠) 𝑄))
5049eqeq2d 2827 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = ((𝑟 𝑠) 𝑄)))
5148, 50anbi12d 618 . . . . . . . . . . 11 (𝑦 = (𝑟 𝑠) → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))))
5251rspcev 3513 . . . . . . . . . 10 (((𝑟 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
5331, 34, 46, 52syl12anc 856 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
54533expia 1143 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
5554expd 402 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑠𝐴 → ((𝑄𝑠𝑧 = (𝑄 𝑠)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
5655rexlimdv 3229 . . . . . 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 1456 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
59 simpr2l 1302 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
60 simpr1 1241 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ¬ 𝑄 𝑧)
61 simpll1 1262 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
6261hllatd 35162 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ Lat)
633, 10llnbase 35307 . . . . . . . . . . . 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 1304 . . . . . . . . . . . 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 17284 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 (𝑧 𝑟))
6862, 64, 66, 67syl3anc 1483 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 (𝑧 𝑟))
69 simpr3r 1308 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑟))
7068, 69breqtrrd 4883 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 𝑋)
71 simplr 776 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑋)
72 simpll3 1266 . . . . . . . . . . 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 1264 . . . . . . . . . . 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 17286 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7762, 64, 73, 75, 76syl13anc 1484 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7870, 71, 77mpbi2and 694 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) 𝑋)
793, 8latjcl 17275 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 𝑄) ∈ (Base‘𝐾))
8062, 64, 73, 79syl3anc 1483 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ (Base‘𝐾))
81 eqid 2817 . . . . . . . . . . . . 13 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
823, 7, 8, 81, 9cvr1 35208 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄𝐴) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8361, 64, 72, 82syl3anc 1483 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8460, 83mpbid 223 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧( ⋖ ‘𝐾)(𝑧 𝑄))
853, 81, 10, 4lplni 35330 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ (Base‘𝐾) ∧ 𝑧𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 𝑄)) → (𝑧 𝑄) ∈ 𝑃)
8661, 80, 59, 84, 85syl31anc 1485 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ 𝑃)
877, 4lplncmp 35360 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ 𝑃𝑋𝑃) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8861, 86, 74, 87syl3anc 1483 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8978, 88mpbid 223 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) = 𝑋)
9089eqcomd 2823 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑄))
91 breq2 4859 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑄 𝑦𝑄 𝑧))
9291notbid 309 . . . . . . . 8 (𝑦 = 𝑧 → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 𝑧))
93 oveq1 6890 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 𝑄) = (𝑧 𝑄))
9493eqeq2d 2827 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = (𝑧 𝑄)))
9592, 94anbi12d 618 . . . . . . 7 (𝑦 = 𝑧 → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))))
9695rspcev 3513 . . . . . 6 ((𝑧𝑁 ∧ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
9759, 60, 90, 96syl12anc 856 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
98973exp2 1456 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (¬ 𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
9958, 98pm2.61d 171 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
10099rexlimdvv 3236 . 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2989  wrex 3108   class class class wbr 4855  cfv 6110  (class class class)co 6883  Basecbs 16087  lecple 16179  joincjn 17168  Latclat 17269  ccvr 35060  Atomscatm 35061  HLchlt 35148  LLinesclln 35289  LPlanesclpl 35290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-riota 6844  df-ov 6886  df-oprab 6887  df-proset 17152  df-poset 17170  df-plt 17182  df-lub 17198  df-glb 17199  df-join 17200  df-meet 17201  df-p0 17263  df-lat 17270  df-clat 17332  df-oposet 34974  df-ol 34976  df-oml 34977  df-covers 35064  df-ats 35065  df-atl 35096  df-cvlat 35120  df-hlat 35149  df-llines 35296  df-lplanes 35297
This theorem is referenced by: (None)
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