Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lubl | Structured version Visualization version GIF version |
Description: The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.) |
Ref | Expression |
---|---|
lublem.b | ⊢ 𝐵 = (Base‘𝐾) |
lublem.l | ⊢ ≤ = (le‘𝐾) |
lublem.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
lubl | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lublem.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lublem.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | lublem.u | . . . 4 ⊢ 𝑈 = (lub‘𝐾) | |
4 | 1, 2, 3 | lublem 17716 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧))) |
5 | 4 | simprd 496 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)) |
6 | breq2 5061 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑋)) | |
7 | 6 | ralbidv 3194 | . . . 4 ⊢ (𝑧 = 𝑋 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋)) |
8 | breq2 5061 | . . . 4 ⊢ (𝑧 = 𝑋 → ((𝑈‘𝑆) ≤ 𝑧 ↔ (𝑈‘𝑆) ≤ 𝑋)) | |
9 | 7, 8 | imbi12d 346 | . . 3 ⊢ (𝑧 = 𝑋 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋))) |
10 | 9 | rspccva 3619 | . 2 ⊢ ((∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧) ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋)) |
11 | 5, 10 | stoic3 1768 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 lubclub 17540 CLatccla 17705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-lub 17572 df-clat 17706 |
This theorem is referenced by: lubss 17719 lubun 17721 |
Copyright terms: Public domain | W3C validator |