Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2lplnmN | Structured version Visualization version GIF version |
Description: If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2lplnm.j | ⊢ ∨ = (join‘𝐾) |
2lplnm.m | ⊢ ∧ = (meet‘𝐾) |
2lplnm.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
2lplnm.n | ⊢ 𝑁 = (LLines‘𝐾) |
2lplnm.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
2lplnmN | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1189 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝑌 ∈ 𝑃) | |
2 | simpl1 1187 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝐾 ∈ HL) | |
3 | hllat 36493 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
4 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 2lplnm.p | . . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾) | |
6 | 4, 5 | lplnbase 36664 | . . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
7 | 4, 5 | lplnbase 36664 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → 𝑌 ∈ (Base‘𝐾)) |
8 | 2lplnm.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
9 | 4, 8 | latmcl 17656 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
10 | 3, 6, 7, 9 | syl3an 1156 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
11 | 10 | adantr 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) |
12 | 7 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑌 ∈ (Base‘𝐾)) |
13 | 12 | adantr 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → 𝑌 ∈ (Base‘𝐾)) |
14 | simp1 1132 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝐾 ∈ HL) | |
15 | 6 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → 𝑋 ∈ (Base‘𝐾)) |
16 | 2lplnm.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
17 | 2lplnm.c | . . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
18 | 4, 16, 8, 17 | cvrexch 36550 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌))) |
19 | 14, 15, 12, 18 | syl3anc 1367 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ((𝑋 ∧ 𝑌)𝐶𝑌 ↔ 𝑋𝐶(𝑋 ∨ 𝑌))) |
20 | 19 | biimpar 480 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌)𝐶𝑌) |
21 | 2lplnm.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
22 | 4, 17, 21, 5 | llncvrlpln 36688 | . . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑋 ∧ 𝑌)𝐶𝑌) → ((𝑋 ∧ 𝑌) ∈ 𝑁 ↔ 𝑌 ∈ 𝑃)) |
23 | 2, 11, 13, 20, 22 | syl31anc 1369 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → ((𝑋 ∧ 𝑌) ∈ 𝑁 ↔ 𝑌 ∈ 𝑃)) |
24 | 1, 23 | mpbird 259 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) ∧ 𝑋𝐶(𝑋 ∨ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 joincjn 17548 meetcmee 17549 Latclat 17649 ⋖ ccvr 36392 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 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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