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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlexch4N | Structured version Visualization version GIF version |
Description: An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.) |
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 |
---|---|
cvlexch4N | ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatl 36951 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ AtLat) |
3 | simpr1 1195 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴) | |
4 | simpr3 1197 | . . . 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 36943 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
11 | 2, 3, 4, 10 | syl3anc 1372 | . . 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 | cvlexchb1 36956 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
14 | 13 | 3expia 1122 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))) |
15 | 11, 14 | sylbird 263 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ∧ 𝑋) = 0 → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))) |
16 | 15 | 3impia 1118 | 1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 class class class wbr 5027 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 lecple 16668 joincjn 17663 meetcmee 17664 0.cp0 17756 Atomscatm 36889 AtLatcal 36890 CvLatclc 36891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-lat 17765 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 |
This theorem is referenced by: hlexch4N 37018 |
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