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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 6103 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾)) | |
2 | isclat.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
3 | 1, 2 | syl6eqr 2662 | . . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈) |
4 | 3 | dmeqd 5248 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈) |
5 | fveq2 6103 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
6 | isclat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 5, 6 | syl6eqr 2662 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
8 | 7 | pweqd 4113 | . . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵) |
9 | 4, 8 | eqeq12d 2625 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵)) |
10 | fveq2 6103 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾)) | |
11 | isclat.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
12 | 10, 11 | syl6eqr 2662 | . . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺) |
13 | 12 | dmeqd 5248 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺) |
14 | 13, 8 | eqeq12d 2625 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵)) |
15 | 9, 14 | anbi12d 743 | . 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 16931 | . 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))} | |
17 | 15, 16 | elrab2 3333 | 1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 Basecbs 15695 Posetcpo 16763 lubclub 16765 glbcglb 16766 CLatccla 16930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-clat 16931 |
This theorem is referenced by: clatpos 16933 clatlem 16934 clatlubcl2 16936 clatglbcl2 16938 clatl 16939 oduclatb 16967 xrsclat 29011 |
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