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Mirrors > Home > MPE Home > Th. List > isclat | Structured version Visualization version GIF version |
Description: The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
isclat.b | ⊢ 𝐵 = (Base‘𝐾) |
isclat.u | ⊢ 𝑈 = (lub‘𝐾) |
isclat.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
isclat | ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6645 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾)) | |
2 | isclat.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
3 | 1, 2 | eqtr4di 2851 | . . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈) |
4 | 3 | dmeqd 5738 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈) |
5 | fveq2 6645 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
6 | isclat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 5, 6 | eqtr4di 2851 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
8 | 7 | pweqd 4516 | . . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵) |
9 | 4, 8 | eqeq12d 2814 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵)) |
10 | fveq2 6645 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾)) | |
11 | isclat.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
12 | 10, 11 | eqtr4di 2851 | . . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺) |
13 | 12 | dmeqd 5738 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺) |
14 | 13, 8 | eqeq12d 2814 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵)) |
15 | 9, 14 | anbi12d 633 | . 2 ⊢ (𝑙 = 𝐾 → ((dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙)) ↔ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
16 | df-clat 17710 | . 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))} | |
17 | 15, 16 | elrab2 3631 | 1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 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 |
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-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-clat 17710 |
This theorem is referenced by: clatpos 17712 clatlem 17713 clatlubcl2 17715 clatglbcl2 17717 clatl 17718 oduclatb 17746 xrsclat 30714 |
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