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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 29214 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 34643 | . 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 34626 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
8 | 1, 7 | syl3an1 1359 | 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 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 joincjn 16944 CLatccla 17107 OMLcoml 34462 ⋖ ccvr 34549 Atomscatm 34550 CvLatclc 34552 HLchlt 34637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 |
This theorem is referenced by: cvr2N 34697 hlrelat3 34698 cvrval3 34699 cvrval4N 34700 cvrexchlem 34705 cvratlem 34707 cvrat3 34728 3dim0 34743 2dim 34756 1cvrjat 34761 llncvrlpln2 34843 lplnexllnN 34850 lplncvrlvol2 34901 lhp2lt 35287 |
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