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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 6877 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾)) | |
2 | isclat.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
3 | 1, 2 | eqtr4di 2789 | . . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈) |
4 | 3 | dmeqd 5896 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈) |
5 | fveq2 6877 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
6 | isclat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 5, 6 | eqtr4di 2789 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
8 | 7 | pweqd 4612 | . . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵) |
9 | 4, 8 | eqeq12d 2747 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵)) |
10 | fveq2 6877 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾)) | |
11 | isclat.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
12 | 10, 11 | eqtr4di 2789 | . . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺) |
13 | 12 | dmeqd 5896 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺) |
14 | 13, 8 | eqeq12d 2747 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵)) |
15 | 9, 14 | anbi12d 631 | . 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 18433 | . 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))} | |
17 | 15, 16 | elrab2 3681 | 1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 𝒫 cpw 4595 dom cdm 5668 ‘cfv 6531 Basecbs 17125 Posetcpo 18241 lubclub 18243 glbcglb 18244 CLatccla 18432 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3474 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5141 df-dm 5678 df-iota 6483 df-fv 6539 df-clat 18433 |
This theorem is referenced by: clatpos 18435 clatlem 18436 clatlubcl2 18438 clatglbcl2 18440 oduclatb 18441 clatl 18442 xrsclat 32049 isclatd 47244 |
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