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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlexch3 | Structured version Visualization version GIF version |
Description: An atomic covering lattice has the exchange property. (atexch 28624 analog.) (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlexch3.b | ⊢ 𝐵 = (Base‘𝐾) |
cvlexch3.l | ⊢ ≤ = (le‘𝐾) |
cvlexch3.j | ⊢ ∨ = (join‘𝐾) |
cvlexch3.m | ⊢ ∧ = (meet‘𝐾) |
cvlexch3.z | ⊢ 0 = (0.‘𝐾) |
cvlexch3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvlexch3 | ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatl 33630 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ AtLat) |
3 | simpr1 1060 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴) | |
4 | simpr3 1062 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
5 | cvlexch3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
6 | cvlexch3.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
7 | cvlexch3.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
8 | cvlexch3.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
9 | cvlexch3.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 5, 6, 7, 8, 9 | atnle 33622 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
11 | 2, 3, 4, 10 | syl3anc 1318 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
12 | cvlexch3.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
13 | 5, 6, 12, 9 | cvlexch1 33633 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
14 | 13 | 3expia 1259 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))) |
15 | 11, 14 | sylbird 249 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ∧ 𝑋) = 0 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))) |
16 | 15 | 3impia 1253 | 1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 0.cp0 16860 Atomscatm 33568 AtLatcal 33569 CvLatclc 33570 |
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-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 |
This theorem is referenced by: hlexch3 33695 |
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