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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 17728 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧))) |
5 | 4 | simprd 498 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)) |
6 | breq2 5070 | . . . . 5 ⊢ (𝑧 = 𝑋 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑋)) | |
7 | 6 | ralbidv 3197 | . . . 4 ⊢ (𝑧 = 𝑋 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋)) |
8 | breq2 5070 | . . . 4 ⊢ (𝑧 = 𝑋 → ((𝑈‘𝑆) ≤ 𝑧 ↔ (𝑈‘𝑆) ≤ 𝑋)) | |
9 | 7, 8 | imbi12d 347 | . . 3 ⊢ (𝑧 = 𝑋 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋))) |
10 | 9 | rspccva 3622 | . 2 ⊢ ((∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧) ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋)) |
11 | 5, 10 | stoic3 1777 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 lubclub 17552 CLatccla 17717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-lub 17584 df-clat 17718 |
This theorem is referenced by: lubss 17731 lubun 17733 |
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