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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2atnelpln | Structured version Visualization version GIF version |
Description: The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.) |
Ref | Expression |
---|---|
2atnelpln.j | ⊢ ∨ = (join‘𝐾) |
2atnelpln.a | ⊢ 𝐴 = (Atoms‘𝐾) |
2atnelpln.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
2atnelpln | ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ¬ (𝑄 ∨ 𝑅) ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 39061 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → 𝐾 ∈ Lat) |
3 | eqid 2726 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 2atnelpln.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | 2atnelpln.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 3, 4, 5 | hlatjcl 39065 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) |
7 | eqid 2726 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 3, 7 | latref 18466 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑅)) |
9 | 2, 6, 8 | syl2anc 582 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑅)) |
10 | simpl1 1188 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∨ 𝑅) ∈ 𝑃) → 𝐾 ∈ HL) | |
11 | simpr 483 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∨ 𝑅) ∈ 𝑃) → (𝑄 ∨ 𝑅) ∈ 𝑃) | |
12 | simpl2 1189 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∨ 𝑅) ∈ 𝑃) → 𝑄 ∈ 𝐴) | |
13 | simpl3 1190 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∨ 𝑅) ∈ 𝑃) → 𝑅 ∈ 𝐴) | |
14 | 2atnelpln.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
15 | 7, 4, 5, 14 | lplnnle2at 39240 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ((𝑄 ∨ 𝑅) ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ¬ (𝑄 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑅)) |
16 | 10, 11, 12, 13, 15 | syl13anc 1369 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∨ 𝑅) ∈ 𝑃) → ¬ (𝑄 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑅)) |
17 | 16 | ex 411 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ((𝑄 ∨ 𝑅) ∈ 𝑃 → ¬ (𝑄 ∨ 𝑅)(le‘𝐾)(𝑄 ∨ 𝑅))) |
18 | 9, 17 | mt2d 136 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ¬ (𝑄 ∨ 𝑅) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 lecple 17273 joincjn 18336 Latclat 18456 Atomscatm 38961 HLchlt 39048 LPlanesclpl 39191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-lat 18457 df-clat 18524 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 |
This theorem is referenced by: islpln2a 39247 |
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