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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 2737 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
| 3 | clatglbcl.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
| 4 | 1, 2, 3 | clatlem 18547 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (((lub‘𝐾)‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
| 5 | 4 | simprd 495 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 lubclub 18355 glbcglb 18356 CLatccla 18543 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-lub 18391 df-glb 18392 df-clat 18544 |
| This theorem is referenced by: clatleglb 18563 clatglbss 18564 clatp0cl 32966 glbconNOLD 39379 pmapglbx 39771 diaglbN 41057 diaintclN 41060 dibglbN 41168 dibintclN 41169 dihglblem2N 41296 dihglblem3N 41297 dihglblem4 41299 dihglbcpreN 41302 dihglblem6 41342 dihintcl 41346 dochval2 41354 dochcl 41355 dochvalr 41359 dochss 41367 |
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