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Mirrors > Home > MPE Home > Th. List > clatglbcl | Structured version Visualization version GIF version |
Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
clatglbcl.b | ⊢ 𝐵 = (Base‘𝐾) |
clatglbcl.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
clatglbcl | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatglbcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | clatglbcl.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
4 | 1, 2, 3 | clatlem 17715 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (((lub‘𝐾)‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
5 | 4 | simprd 498 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ‘cfv 6350 Basecbs 16477 lubclub 17546 glbcglb 17547 CLatccla 17711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-lub 17578 df-glb 17579 df-clat 17712 |
This theorem is referenced by: clatleglb 17730 clatglbss 17731 clatp0cl 30653 glbconN 36507 pmapglbx 36899 diaglbN 38185 diaintclN 38188 dibglbN 38296 dibintclN 38297 dihglblem2N 38424 dihglblem3N 38425 dihglblem4 38427 dihglbcpreN 38430 dihglblem6 38470 dihintcl 38474 dochval2 38482 dochcl 38483 dochvalr 38487 dochss 38495 |
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