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Mathbox for Norm Megill |
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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 31471 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 38029 | . 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 38012 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
8 | 1, 7 | syl3an1 1163 | 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 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 lecple 17186 joincjn 18246 CLatccla 18433 OMLcoml 37848 ⋖ ccvr 37935 Atomscatm 37936 CvLatclc 37938 HLchlt 38023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-lat 18367 df-clat 18434 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 |
This theorem is referenced by: cvr2N 38085 hlrelat3 38086 cvrval3 38087 cvrval4N 38088 cvrexchlem 38093 cvratlem 38095 cvrat3 38116 3dim0 38131 2dim 38144 1cvrjat 38149 llncvrlpln2 38231 lplnexllnN 38238 lplncvrlvol2 38289 lhp2lt 38675 |
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