Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 30433 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 37105 | . 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 37088 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
8 | 1, 7 | syl3an1 1165 | 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 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5050 ‘cfv 6377 (class class class)co 7210 Basecbs 16757 lecple 16806 joincjn 17815 CLatccla 18001 OMLcoml 36924 ⋖ ccvr 37011 Atomscatm 37012 CvLatclc 37014 HLchlt 37099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 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 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-proset 17799 df-poset 17817 df-plt 17833 df-lub 17849 df-glb 17850 df-join 17851 df-meet 17852 df-p0 17928 df-lat 17935 df-clat 18002 df-oposet 36925 df-ol 36927 df-oml 36928 df-covers 37015 df-ats 37016 df-atl 37047 df-cvlat 37071 df-hlat 37100 |
This theorem is referenced by: cvr2N 37160 hlrelat3 37161 cvrval3 37162 cvrval4N 37163 cvrexchlem 37168 cvratlem 37170 cvrat3 37191 3dim0 37206 2dim 37219 1cvrjat 37224 llncvrlpln2 37306 lplnexllnN 37313 lplncvrlvol2 37364 lhp2lt 37750 |
Copyright terms: Public domain | W3C validator |