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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlexchb2 | Structured version Visualization version GIF version |
Description: An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.) |
Ref | Expression |
---|---|
cvlexch.b | ⊢ 𝐵 = (Base‘𝐾) |
cvlexch.l | ⊢ ≤ = (le‘𝐾) |
cvlexch.j | ⊢ ∨ = (join‘𝐾) |
cvlexch.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvlexchb2 | ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlexch.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cvlexch.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | cvlexch.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | cvlexch.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | cvlexchb1 39309 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
6 | cvllat 39305 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat) | |
7 | 6 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝐾 ∈ Lat) |
8 | simp22 1208 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑄 ∈ 𝐴) | |
9 | 1, 4 | atbase 39268 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑄 ∈ 𝐵) |
11 | simp23 1209 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
12 | 1, 3 | latjcom 18488 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄)) |
13 | 7, 10, 11, 12 | syl3anc 1373 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄)) |
14 | 13 | breq2d 5153 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄))) |
15 | simp21 1207 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) | |
16 | 1, 4 | atbase 39268 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
18 | 1, 3 | latjcom 18488 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∨ 𝑋) = (𝑋 ∨ 𝑃)) |
19 | 7, 17, 11, 18 | syl3anc 1373 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ∨ 𝑋) = (𝑋 ∨ 𝑃)) |
20 | 19, 13 | eqeq12d 2752 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))) |
21 | 5, 14, 20 | 3bitr4d 311 | 1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5141 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 lecple 17300 joincjn 18353 Latclat 18472 Atomscatm 39242 CvLatclc 39244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-proset 18336 df-poset 18355 df-lub 18387 df-glb 18388 df-join 18389 df-meet 18390 df-lat 18473 df-ats 39246 df-atl 39277 df-cvlat 39301 |
This theorem is referenced by: hlexchb2 39365 |
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