Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clatglb | Structured version Visualization version GIF version |
Description: Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.) |
Ref | Expression |
---|---|
clatglb.b | ⊢ 𝐵 = (Base‘𝐾) |
clatglb.l | ⊢ ≤ = (le‘𝐾) |
clatglb.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
clatglb | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 (𝐺‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝐺‘𝑆)))) |
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 485 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat) | |
5 | 1, 3 | clatglbcl2 17719 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
6 | 1, 2, 3, 4, 5 | glbprop 17603 | 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 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 lecple 16566 glbcglb 17547 CLatccla 17711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-glb 17579 df-clat 17712 |
This theorem is referenced by: clatleglb 17730 |
Copyright terms: Public domain | W3C validator |