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| 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 2736 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 4 | 1, 2, 3 | clatlem 18548 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ ((glb‘𝐾)‘𝑆) ∈ 𝐵)) | 
| 5 | 4 | simpld 494 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 Basecbs 17248 lubclub 18356 glbcglb 18357 CLatccla 18544 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-lub 18392 df-glb 18393 df-clat 18545 | 
| This theorem is referenced by: oduclatb 18553 lubss 18559 lubun 18561 clatp1cl 32968 atlatmstc 39321 polsubN 39910 2polvalN 39917 2polssN 39918 3polN 39919 2pmaplubN 39929 paddunN 39930 poldmj1N 39931 pnonsingN 39936 ispsubcl2N 39950 psubclinN 39951 paddatclN 39952 polsubclN 39955 poml4N 39956 | 
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