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Theorem clatglb 18473

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 483	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat)
5	1, 3	clatglbcl2 18463	. 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)
6	1, 2, 3, 4, 5	glbprop 18326	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 396 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ⊆ wss 3883 class class class wbr 5072 ‘cfv 6485 Basecbs 17170 lecple 17218 glbcglb 18267 CLatccla 18455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-glb 18302 df-clat 18456

This theorem is referenced by: clatleglb 18475

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