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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvr1 | Structured version Visualization version GIF version |
Description: A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 30131 analog.) (Contributed by NM, 17-Nov-2011.) |
Ref | Expression |
---|---|
cvr1.b | ⊢ 𝐵 = (Base‘𝐾) |
cvr1.l | ⊢ ≤ = (le‘𝐾) |
cvr1.j | ⊢ ∨ = (join‘𝐾) |
cvr1.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
cvr1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvr1 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmcv 36491 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | cvr1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cvr1.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | cvr1.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
5 | cvr1.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
6 | cvr1.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 2, 3, 4, 5, 6 | cvlcvr1 36474 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
8 | 1, 7 | syl3an1 1159 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 lecple 16571 joincjn 17553 CLatccla 17716 OMLcoml 36310 ⋖ ccvr 36397 Atomscatm 36398 CvLatclc 36400 HLchlt 36485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-lat 17655 df-clat 17717 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 |
This theorem is referenced by: cvr2N 36546 hlrelat3 36547 cvrval3 36548 cvrval4N 36549 cvrexchlem 36554 cvratlem 36556 cvrat3 36577 3dim0 36592 2dim 36605 1cvrjat 36610 llncvrlpln2 36692 lplnexllnN 36699 lplncvrlvol2 36750 lhp2lt 37136 |
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