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Mirrors > Home > MPE Home > Th. List > clatlem | Structured version Visualization version GIF version |
Description: Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
clatlem.b | ⊢ 𝐵 = (Base‘𝐾) |
clatlem.u | ⊢ 𝑈 = (lub‘𝐾) |
clatlem.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
clatlem | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatlem.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | clatlem.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
3 | simpl 486 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat) | |
4 | 1 | fvexi 6659 | . . . . . . 7 ⊢ 𝐵 ∈ V |
5 | 4 | elpw2 5212 | . . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
6 | 5 | biimpri 231 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
7 | 6 | adantl 485 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
8 | clatlem.g | . . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾) | |
9 | 1, 2, 8 | isclat 17711 | . . . . . . 7 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
10 | 9 | biimpi 219 | . . . . . 6 ⊢ (𝐾 ∈ CLat → (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
12 | 11 | simprld 771 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝑈 = 𝒫 𝐵) |
13 | 7, 12 | eleqtrrd 2893 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) |
14 | 1, 2, 3, 13 | lubcl 17587 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
15 | 11 | simprrd 773 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
16 | 7, 15 | eleqtrrd 2893 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
17 | 1, 8, 3, 16 | glbcl 17600 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
18 | 14, 17 | jca 515 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 dom cdm 5519 ‘cfv 6324 Basecbs 16475 Posetcpo 17542 lubclub 17544 glbcglb 17545 CLatccla 17709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-lub 17576 df-glb 17577 df-clat 17710 |
This theorem is referenced by: clatlubcl 17714 clatglbcl 17716 |
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