Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnexatN | Structured version Visualization version GIF version |
Description: Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.) |
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) |
Ref | Expression |
---|---|
lplnexatN | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ 𝑋 = (𝑌 ∨ 𝑞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → 𝐾 ∈ HL) | |
2 | simp3 1134 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → 𝑌 ∈ 𝑁) | |
3 | simp2 1133 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → 𝑋 ∈ 𝑃) | |
4 | 1, 2, 3 | 3jca 1124 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → (𝐾 ∈ HL ∧ 𝑌 ∈ 𝑁 ∧ 𝑋 ∈ 𝑃)) |
5 | lplnexat.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
6 | eqid 2821 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
7 | lplnexat.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
8 | lplnexat.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
9 | 5, 6, 7, 8 | llncvrlpln2 36687 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑁 ∧ 𝑋 ∈ 𝑃) ∧ 𝑌 ≤ 𝑋) → 𝑌( ⋖ ‘𝐾)𝑋) |
10 | 4, 9 | sylan 582 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → 𝑌( ⋖ ‘𝐾)𝑋) |
11 | simpl1 1187 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ HL) | |
12 | simpl3 1189 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → 𝑌 ∈ 𝑁) | |
13 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 7 | llnbase 36639 | . . . . 5 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾)) |
15 | 12, 14 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → 𝑌 ∈ (Base‘𝐾)) |
16 | simpl2 1188 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → 𝑋 ∈ 𝑃) | |
17 | 13, 8 | lplnbase 36664 | . . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → 𝑋 ∈ (Base‘𝐾)) |
19 | lplnexat.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
20 | lplnexat.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
21 | 13, 5, 19, 6, 20 | cvrval3 36543 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑌( ⋖ ‘𝐾)𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ (𝑌 ∨ 𝑞) = 𝑋))) |
22 | 11, 15, 18, 21 | syl3anc 1367 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → (𝑌( ⋖ ‘𝐾)𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ (𝑌 ∨ 𝑞) = 𝑋))) |
23 | eqcom 2828 | . . . . 5 ⊢ ((𝑌 ∨ 𝑞) = 𝑋 ↔ 𝑋 = (𝑌 ∨ 𝑞)) | |
24 | 23 | anbi2i 624 | . . . 4 ⊢ ((¬ 𝑞 ≤ 𝑌 ∧ (𝑌 ∨ 𝑞) = 𝑋) ↔ (¬ 𝑞 ≤ 𝑌 ∧ 𝑋 = (𝑌 ∨ 𝑞))) |
25 | 24 | rexbii 3247 | . . 3 ⊢ (∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ (𝑌 ∨ 𝑞) = 𝑋) ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ 𝑋 = (𝑌 ∨ 𝑞))) |
26 | 22, 25 | syl6bb 289 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → (𝑌( ⋖ ‘𝐾)𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ 𝑋 = (𝑌 ∨ 𝑞)))) |
27 | 10, 26 | mpbid 234 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ 𝑌 ≤ 𝑋) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑌 ∧ 𝑋 = (𝑌 ∨ 𝑞))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 ⋖ ccvr 36392 Atomscatm 36393 HLchlt 36480 LLinesclln 36621 LPlanesclpl 36622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 |
This theorem is referenced by: (None) |
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