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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatmstcOLDN | Structured version Visualization version GIF version |
Description: An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 32385 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlatmstc.b | ⊢ 𝐵 = (Base‘𝐾) |
hlatmstc.l | ⊢ ≤ = (le‘𝐾) |
hlatmstc.u | ⊢ 𝑈 = (lub‘𝐾) |
hlatmstc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatmstcOLDN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmat 39270 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
2 | hlatmstc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | hlatmstc.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | hlatmstc.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
5 | hlatmstc.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 2, 3, 4, 5 | atlatmstc 39224 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋) |
7 | 1, 6 | sylan 579 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 {crab 3438 class class class wbr 5169 ‘cfv 6572 Basecbs 17253 lecple 17313 lubclub 18374 CLatccla 18563 OMLcoml 39080 Atomscatm 39168 AtLatcal 39169 HLchlt 39255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-proset 18360 df-poset 18378 df-plt 18395 df-lub 18411 df-glb 18412 df-join 18413 df-meet 18414 df-p0 18490 df-lat 18497 df-clat 18564 df-oposet 39081 df-ol 39083 df-oml 39084 df-covers 39171 df-ats 39172 df-atl 39203 df-cvlat 39227 df-hlat 39256 |
This theorem is referenced by: (None) |
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