clatglbcl - Metamath Proof Explorer

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Theorem clatglbcl 18496

Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011.)

**Hypotheses**
Ref	Expression
clatglbcl.b	⊢ 𝐵 = (Base‘𝐾)
clatglbcl.g	⊢ 𝐺 = (glb‘𝐾)

**Assertion**
Ref	Expression
clatglbcl	⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵)

**Proof of Theorem clatglbcl**
Step	Hyp	Ref	Expression
1		clatglbcl.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2728	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		clatglbcl.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4	1, 2, 3	clatlem 18493	. 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 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6548 Basecbs 17179 lubclub 18300 glbcglb 18301 CLatccla 18489

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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-lub 18337 df-glb 18338 df-clat 18490

This theorem is referenced by: clatleglb 18509 clatglbss 18510 clatp0cl 32703 glbconNOLD 38850 pmapglbx 39242 diaglbN 40528 diaintclN 40531 dibglbN 40639 dibintclN 40640 dihglblem2N 40767 dihglblem3N 40768 dihglblem4 40770 dihglbcpreN 40773 dihglblem6 40813 dihintcl 40817 dochval2 40825 dochcl 40826 dochvalr 40830 dochss 40838

	
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