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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 2726 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | clatglbcl.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
4 | 1, 2, 3 | clatlem 18527 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (((lub‘𝐾)‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
5 | 4 | simprd 494 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6554 Basecbs 17213 lubclub 18334 glbcglb 18335 CLatccla 18523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-lub 18371 df-glb 18372 df-clat 18524 |
This theorem is referenced by: clatleglb 18543 clatglbss 18544 clatp0cl 32846 glbconNOLD 39076 pmapglbx 39468 diaglbN 40754 diaintclN 40757 dibglbN 40865 dibintclN 40866 dihglblem2N 40993 dihglblem3N 40994 dihglblem4 40996 dihglbcpreN 40999 dihglblem6 41039 dihintcl 41043 dochval2 41051 dochcl 41052 dochvalr 41056 dochss 41064 |
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