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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 30150 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 36453 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ AtLat) |
3 | simpr1 1189 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴) | |
4 | simpr3 1191 | . . . 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 36445 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) |
11 | 2, 3, 4, 10 | syl3anc 1366 | . . 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 36456 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))) |
14 | 13 | 3expia 1116 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))) |
15 | 11, 14 | sylbird 262 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ∧ 𝑋) = 0 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))) |
16 | 15 | 3impia 1112 | 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 208 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 lecple 16564 joincjn 17546 meetcmee 17547 0.cp0 17639 Atomscatm 36391 AtLatcal 36392 CvLatclc 36393 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-lat 17648 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 |
This theorem is referenced by: hlexch3 36519 |
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