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Theorem lplnexllnN 33868
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 1058 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝑋𝑃)
2 simpl1 1057 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝐾 ∈ HL)
3 eqid 2610 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
53, 4lplnbase 33838 . . . . 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 33837 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
122, 6, 11syl2anc 691 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
131, 12mpbid 221 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)))
14 simpll1 1093 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
15 simpr2l 1113 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
16 simpll3 1095 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
17 simpr1 1060 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑧)
187, 8, 9, 10llnexatN 33825 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧𝑁𝑄𝐴) ∧ 𝑄 𝑧) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
1914, 15, 16, 17, 18syl31anc 1321 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
20 simp1l1 1147 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ HL)
21 simp22r 1174 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝐴)
22 simp3l 1082 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠𝐴)
23 simp1l3 1149 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝐴)
24 simp23l 1175 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 𝑧)
25 simp3rr 1128 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑧 = (𝑄 𝑠))
2625breq2d 4595 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑧𝑟 (𝑄 𝑠)))
2724, 26mtbid 313 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 (𝑄 𝑠))
287, 8, 9atnlej2 33684 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑠𝐴) ∧ ¬ 𝑟 (𝑄 𝑠)) → 𝑟𝑠)
2920, 21, 23, 22, 27, 28syl131anc 1331 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝑠)
308, 9, 10llni2 33816 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) ∧ 𝑟𝑠) → (𝑟 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1321 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑠) ∈ 𝑁)
32 simp3rl 1127 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝑠)
337, 8, 9hlatcon2 33756 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑠𝐴𝑟𝐴) ∧ (𝑄𝑠 ∧ ¬ 𝑟 (𝑄 𝑠))) → ¬ 𝑄 (𝑟 𝑠))
3420, 23, 22, 21, 32, 27, 33syl132anc 1336 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑄 (𝑟 𝑠))
35 simp23r 1176 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = (𝑧 𝑟))
3625oveq1d 6564 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑧 𝑟) = ((𝑄 𝑠) 𝑟))
37 hllat 33668 . . . . . . . . . . . . 13 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3820, 37syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ Lat)
393, 9atbase 33594 . . . . . . . . . . . . 13 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4023, 39syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄 ∈ (Base‘𝐾))
413, 9atbase 33594 . . . . . . . . . . . . 13 (𝑠𝐴𝑠 ∈ (Base‘𝐾))
4222, 41syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠 ∈ (Base‘𝐾))
433, 9atbase 33594 . . . . . . . . . . . . 13 (𝑟𝐴𝑟 ∈ (Base‘𝐾))
4421, 43syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟 ∈ (Base‘𝐾))
453, 8latj31 16922 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4638, 40, 42, 44, 45syl13anc 1320 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4735, 36, 463eqtrd 2648 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = ((𝑟 𝑠) 𝑄))
48 breq2 4587 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑄 𝑦𝑄 (𝑟 𝑠)))
4948notbid 307 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 (𝑟 𝑠)))
50 oveq1 6556 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑦 𝑄) = ((𝑟 𝑠) 𝑄))
5150eqeq2d 2620 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = ((𝑟 𝑠) 𝑄)))
5249, 51anbi12d 743 . . . . . . . . . . 11 (𝑦 = (𝑟 𝑠) → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))))
5352rspcev 3282 . . . . . . . . . 10 (((𝑟 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
5431, 34, 47, 53syl12anc 1316 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
55543expia 1259 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
5655expd 451 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑠𝐴 → ((𝑄𝑠𝑧 = (𝑄 𝑠)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
5756rexlimdv 3012 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
5819, 57mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
59583exp2 1277 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
60 simpr2l 1113 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
61 simpr1 1060 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ¬ 𝑄 𝑧)
62 simpll1 1093 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
6362, 37syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ Lat)
643, 10llnbase 33813 . . . . . . . . . . . 12 (𝑧𝑁𝑧 ∈ (Base‘𝐾))
6560, 64syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 ∈ (Base‘𝐾))
66 simpr2r 1114 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑟𝐴)
6766, 43syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑟 ∈ (Base‘𝐾))
683, 7, 8latlej1 16883 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 (𝑧 𝑟))
6963, 65, 67, 68syl3anc 1318 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 (𝑧 𝑟))
70 simpr3r 1116 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑟))
7169, 70breqtrrd 4611 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 𝑋)
72 simplr 788 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑋)
73 simpll3 1095 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
7473, 39syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 ∈ (Base‘𝐾))
75 simpll2 1094 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋𝑃)
7675, 5syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 ∈ (Base‘𝐾))
773, 7, 8latjle12 16885 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7863, 65, 74, 76, 77syl13anc 1320 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7971, 72, 78mpbi2and 958 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) 𝑋)
803, 8latjcl 16874 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 𝑄) ∈ (Base‘𝐾))
8163, 65, 74, 80syl3anc 1318 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ (Base‘𝐾))
82 eqid 2610 . . . . . . . . . . . . 13 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
833, 7, 8, 82, 9cvr1 33714 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄𝐴) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8462, 65, 73, 83syl3anc 1318 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8561, 84mpbid 221 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧( ⋖ ‘𝐾)(𝑧 𝑄))
863, 82, 10, 4lplni 33836 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ (Base‘𝐾) ∧ 𝑧𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 𝑄)) → (𝑧 𝑄) ∈ 𝑃)
8762, 81, 60, 85, 86syl31anc 1321 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ 𝑃)
887, 4lplncmp 33866 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ 𝑃𝑋𝑃) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8962, 87, 75, 88syl3anc 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
9079, 89mpbid 221 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) = 𝑋)
9190eqcomd 2616 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑄))
92 breq2 4587 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑄 𝑦𝑄 𝑧))
9392notbid 307 . . . . . . . 8 (𝑦 = 𝑧 → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 𝑧))
94 oveq1 6556 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 𝑄) = (𝑧 𝑄))
9594eqeq2d 2620 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = (𝑧 𝑄)))
9693, 95anbi12d 743 . . . . . . 7 (𝑦 = 𝑧 → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))))
9796rspcev 3282 . . . . . 6 ((𝑧𝑁 ∧ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
9860, 61, 91, 97syl12anc 1316 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
99983exp2 1277 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (¬ 𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
10059, 99pm2.61d 169 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
101100rexlimdvv 3019 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))
10213, 101mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  ccvr 33567  Atomscatm 33568  HLchlt 33655  LLinesclln 33795  LPlanesclpl 33796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803
This theorem is referenced by: (None)
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