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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatle | Structured version Visualization version GIF version |
Description: The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 29785 analog.) (Contributed by NM, 4-Nov-2011.) |
Ref | Expression |
---|---|
hlatle.b | ⊢ 𝐵 = (Base‘𝐾) |
hlatle.l | ⊢ ≤ = (le‘𝐾) |
hlatle.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatle | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmat 35440 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
2 | hlatle.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | hlatle.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | hlatle.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 2, 3, 4 | atlatle 35395 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))) |
6 | 1, 5 | syl3an1 1208 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 class class class wbr 4873 ‘cfv 6123 Basecbs 16222 lecple 16312 CLatccla 17460 OMLcoml 35250 Atomscatm 35338 AtLatcal 35339 HLchlt 35425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-lat 17399 df-clat 17461 df-oposet 35251 df-ol 35253 df-oml 35254 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 |
This theorem is referenced by: hlateq 35474 trlord 36644 |
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