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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 2797 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | 1, 2, 3 | clatlem 17423 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ ((glb‘𝐾)‘𝑆) ∈ 𝐵)) |
5 | 4 | simpld 489 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3767 ‘cfv 6099 Basecbs 16181 lubclub 17254 glbcglb 17255 CLatccla 17419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-lub 17286 df-glb 17287 df-clat 17420 |
This theorem is referenced by: lubss 17433 lubun 17435 oduclatb 17456 clatp1cl 30180 atlatmstc 35332 polsubN 35920 2polvalN 35927 2polssN 35928 3polN 35929 2pmaplubN 35939 paddunN 35940 poldmj1N 35941 pnonsingN 35946 ispsubcl2N 35960 psubclinN 35961 paddatclN 35962 polsubclN 35965 poml4N 35966 |
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