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

Theorem lplnexllnN 35168
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 1085 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝑋𝑃)
2 simpl1 1084 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → 𝐾 ∈ HL)
3 eqid 2651 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
4 lplnexat.p . . . . . 6 𝑃 = (LPlanes‘𝐾)
53, 4lplnbase 35138 . . . . 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 35137 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
122, 6, 11syl2anc 694 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑋𝑃 ↔ ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟))))
131, 12mpbid 222 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ∃𝑧𝑁𝑟𝐴𝑟 𝑧𝑋 = (𝑧 𝑟)))
14 simpll1 1120 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝐾 ∈ HL)
15 simpr2l 1140 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
16 simpll3 1122 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄𝐴)
17 simpr1 1087 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑧)
187, 8, 9, 10llnexatN 35125 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑧𝑁𝑄𝐴) ∧ 𝑄 𝑧) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
1914, 15, 16, 17, 18syl31anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑠𝐴 (𝑄𝑠𝑧 = (𝑄 𝑠)))
20 simp1l1 1174 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝐾 ∈ HL)
21 simp22r 1201 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝐴)
22 simp3l 1109 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑠𝐴)
23 simp1l3 1176 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝐴)
24 simp23l 1202 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑟 𝑧)
25 simp3rr 1155 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑧 = (𝑄 𝑠))
2625breq2d 4697 . . . . . . . . . . . . 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 34984 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑠𝐴) ∧ ¬ 𝑟 (𝑄 𝑠)) → 𝑟𝑠)
2920, 21, 23, 22, 27, 28syl131anc 1379 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑟𝑠)
308, 9, 10llni2 35116 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) ∧ 𝑟𝑠) → (𝑟 𝑠) ∈ 𝑁)
3120, 21, 22, 29, 30syl31anc 1369 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑟 𝑠) ∈ 𝑁)
32 simp3rl 1154 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑄𝑠)
337, 8, 9hlatcon2 35056 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑠𝐴𝑟𝐴) ∧ (𝑄𝑠 ∧ ¬ 𝑟 (𝑄 𝑠))) → ¬ 𝑄 (𝑟 𝑠))
3420, 23, 22, 21, 32, 27, 33syl132anc 1384 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ¬ 𝑄 (𝑟 𝑠))
35 simp23r 1203 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = (𝑧 𝑟))
3625oveq1d 6705 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → (𝑧 𝑟) = ((𝑄 𝑠) 𝑟))
37 hllat 34968 . . . . . . . . . . . . 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 34894 . . . . . . . . . . . . 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 34894 . . . . . . . . . . . . 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 34894 . . . . . . . . . . . . 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 17146 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾))) → ((𝑄 𝑠) 𝑟) = ((𝑟 𝑠) 𝑄))
4638, 40, 42, 44, 45syl13anc 1368 . . . . . . . . . . 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 2689 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → 𝑋 = ((𝑟 𝑠) 𝑄))
48 breq2 4689 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑄 𝑦𝑄 (𝑟 𝑠)))
4948notbid 307 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 (𝑟 𝑠)))
50 oveq1 6697 . . . . . . . . . . . . 13 (𝑦 = (𝑟 𝑠) → (𝑦 𝑄) = ((𝑟 𝑠) 𝑄))
5150eqeq2d 2661 . . . . . . . . . . . 12 (𝑦 = (𝑟 𝑠) → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = ((𝑟 𝑠) 𝑄)))
5249, 51anbi12d 747 . . . . . . . . . . 11 (𝑦 = (𝑟 𝑠) → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))))
5352rspcev 3340 . . . . . . . . . 10 (((𝑟 𝑠) ∈ 𝑁 ∧ (¬ 𝑄 (𝑟 𝑠) ∧ 𝑋 = ((𝑟 𝑠) 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
5431, 34, 47, 53syl12anc 1364 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟))) ∧ (𝑠𝐴 ∧ (𝑄𝑠𝑧 = (𝑄 𝑠)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
55543expia 1286 . . . . . . . 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 3059 . . . . . 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 1307 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
60 simpr2l 1140 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧𝑁)
61 simpr1 1087 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ¬ 𝑄 𝑧)
62 simpll1 1120 . . . . . . . . . . . 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 35113 . . . . . . . . . . . 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 1141 . . . . . . . . . . . 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 17107 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → 𝑧 (𝑧 𝑟))
6963, 65, 67, 68syl3anc 1366 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 (𝑧 𝑟))
70 simpr3r 1143 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑟))
7169, 70breqtrrd 4713 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧 𝑋)
72 simplr 807 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑄 𝑋)
73 simpll3 1122 . . . . . . . . . . 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 1121 . . . . . . . . . . 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 17109 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7863, 65, 74, 76, 77syl13anc 1368 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑋𝑄 𝑋) ↔ (𝑧 𝑄) 𝑋))
7971, 72, 78mpbi2and 976 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) 𝑋)
803, 8latjcl 17098 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑧 𝑄) ∈ (Base‘𝐾))
8163, 65, 74, 80syl3anc 1366 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ (Base‘𝐾))
82 eqid 2651 . . . . . . . . . . . . 13 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
833, 7, 8, 82, 9cvr1 35014 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑧 ∈ (Base‘𝐾) ∧ 𝑄𝐴) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8462, 65, 73, 83syl3anc 1366 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (¬ 𝑄 𝑧𝑧( ⋖ ‘𝐾)(𝑧 𝑄)))
8561, 84mpbid 222 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑧( ⋖ ‘𝐾)(𝑧 𝑄))
863, 82, 10, 4lplni 35136 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ (Base‘𝐾) ∧ 𝑧𝑁) ∧ 𝑧( ⋖ ‘𝐾)(𝑧 𝑄)) → (𝑧 𝑄) ∈ 𝑃)
8762, 81, 60, 85, 86syl31anc 1369 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) ∈ 𝑃)
887, 4lplncmp 35166 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑧 𝑄) ∈ 𝑃𝑋𝑃) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
8962, 87, 75, 88syl3anc 1366 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ((𝑧 𝑄) 𝑋 ↔ (𝑧 𝑄) = 𝑋))
9079, 89mpbid 222 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → (𝑧 𝑄) = 𝑋)
9190eqcomd 2657 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → 𝑋 = (𝑧 𝑄))
92 breq2 4689 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑄 𝑦𝑄 𝑧))
9392notbid 307 . . . . . . . 8 (𝑦 = 𝑧 → (¬ 𝑄 𝑦 ↔ ¬ 𝑄 𝑧))
94 oveq1 6697 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦 𝑄) = (𝑧 𝑄))
9594eqeq2d 2661 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋 = (𝑦 𝑄) ↔ 𝑋 = (𝑧 𝑄)))
9693, 95anbi12d 747 . . . . . . 7 (𝑦 = 𝑧 → ((¬ 𝑄 𝑦𝑋 = (𝑦 𝑄)) ↔ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))))
9796rspcev 3340 . . . . . 6 ((𝑧𝑁 ∧ (¬ 𝑄 𝑧𝑋 = (𝑧 𝑄))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
9860, 61, 91, 97syl12anc 1364 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) ∧ (¬ 𝑄 𝑧 ∧ (𝑧𝑁𝑟𝐴) ∧ (¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)))) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))
99983exp2 1307 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → (¬ 𝑄 𝑧 → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄))))))
10059, 99pm2.61d 170 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑃𝑄𝐴) ∧ 𝑄 𝑋) → ((𝑧𝑁𝑟𝐴) → ((¬ 𝑟 𝑧𝑋 = (𝑧 𝑟)) → ∃𝑦𝑁𝑄 𝑦𝑋 = (𝑦 𝑄)))))
101100rexlimdvv 3066 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  ccvr 34867  Atomscatm 34868  HLchlt 34955  LLinesclln 35095  LPlanesclpl 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator