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Theorem clatglb 18567

Description: Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)

**Hypotheses**
Ref	Expression
clatglb.b	⊢ 𝐵 = (Base‘𝐾)
clatglb.l	⊢ ≤ = (le‘𝐾)
clatglb.g	⊢ 𝐺 = (glb‘𝐾)

**Assertion**
Ref	Expression
clatglb	⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 (𝐺‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝐺‘𝑆))))

Distinct variable groups: 𝑦,𝑧,𝐵 𝑦,𝐺,𝑧 𝑦,𝐾,𝑧 𝑦, ≤ ,𝑧 𝑦,𝑆,𝑧

**Proof of Theorem clatglb**
Step	Hyp	Ref	Expression
1		clatglb.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		clatglb.l	. 2 ⊢ ≤ = (le‘𝐾)
3		clatglb.g	. 2 ⊢ 𝐺 = (glb‘𝐾)
4		simpl 487	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat)
5	1, 3	clatglbcl2 18557	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)
6	1, 2, 3, 4, 5	glbprop 18420	1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 (𝐺‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝐺‘𝑆))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ⊆ wss 3905 class class class wbr 5109 ‘cfv 6536 Basecbs 17264 lecple 17312 glbcglb 18361 CLatccla 18549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-glb 18396 df-clat 18550

This theorem is referenced by: clatleglb 18569

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