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Mirrors > Home > MPE Home > Th. List > clatlubcl | Structured version Visualization version GIF version |
Description: Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
clatlubcl.b | ⊢ 𝐵 = (Base‘𝐾) |
clatlubcl.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
clatlubcl | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatlubcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | clatlubcl.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
3 | eqid 2725 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | 1, 2, 3 | clatlem 18497 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ ((glb‘𝐾)‘𝑆) ∈ 𝐵)) |
5 | 4 | simpld 493 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ‘cfv 6549 Basecbs 17183 lubclub 18304 glbcglb 18305 CLatccla 18493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-lub 18341 df-glb 18342 df-clat 18494 |
This theorem is referenced by: oduclatb 18502 lubss 18508 lubun 18510 clatp1cl 32793 atlatmstc 38921 polsubN 39510 2polvalN 39517 2polssN 39518 3polN 39519 2pmaplubN 39529 paddunN 39530 poldmj1N 39531 pnonsingN 39536 ispsubcl2N 39550 psubclinN 39551 paddatclN 39552 polsubclN 39555 poml4N 39556 |
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