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Mirrors > Home > MPE Home > Th. List > lublem | Structured version Visualization version GIF version |
Description: Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.) |
Ref | Expression |
---|---|
lublem.b | ⊢ 𝐵 = (Base‘𝐾) |
lublem.l | ⊢ ≤ = (le‘𝐾) |
lublem.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
lublem | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lublem.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lublem.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | lublem.u | . 2 ⊢ 𝑈 = (lub‘𝐾) | |
4 | simpl 483 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat) | |
5 | 1, 3 | clatlubcl2 18211 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) |
6 | 1, 2, 3, 4, 5 | lubprop 18065 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3888 class class class wbr 5075 ‘cfv 6428 Basecbs 16901 lecple 16958 lubclub 18016 CLatccla 18205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5486 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-lub 18053 df-clat 18206 |
This theorem is referenced by: lubub 18218 lubl 18219 |
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