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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 2622 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
4 | 1, 2, 3 | clatlem 17111 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ ((glb‘𝐾)‘𝑆) ∈ 𝐵)) |
5 | 4 | simpld 475 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 Basecbs 15857 lubclub 16942 glbcglb 16943 CLatccla 17107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-lub 16974 df-glb 16975 df-clat 17108 |
This theorem is referenced by: lubss 17121 lubun 17123 oduclatb 17144 clatp1cl 29672 atlatmstc 34606 polsubN 35193 2polvalN 35200 2polssN 35201 3polN 35202 2pmaplubN 35212 paddunN 35213 poldmj1N 35214 pnonsingN 35219 ispsubcl2N 35233 psubclinN 35234 paddatclN 35235 polsubclN 35238 poml4N 35239 |
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