| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 6906 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾)) | |
| 2 | isclat.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
| 3 | 1, 2 | eqtr4di 2795 | . . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈) |
| 4 | 3 | dmeqd 5916 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈) |
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
| 6 | isclat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
| 8 | 7 | pweqd 4617 | . . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵) |
| 9 | 4, 8 | eqeq12d 2753 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵)) |
| 10 | fveq2 6906 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾)) | |
| 11 | isclat.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
| 12 | 10, 11 | eqtr4di 2795 | . . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺) |
| 13 | 12 | dmeqd 5916 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺) |
| 14 | 13, 8 | eqeq12d 2753 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵)) |
| 15 | 9, 14 | anbi12d 632 | . 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 18544 | . 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))} | |
| 17 | 15, 16 | elrab2 3695 | 1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 𝒫 cpw 4600 dom cdm 5685 ‘cfv 6561 Basecbs 17247 Posetcpo 18353 lubclub 18355 glbcglb 18356 CLatccla 18543 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-clat 18544 |
| This theorem is referenced by: clatpos 18546 clatlem 18547 clatlubcl2 18549 clatglbcl2 18551 oduclatb 18552 clatl 18553 xrsclat 33013 isclatd 48872 |
| Copyright terms: Public domain | W3C validator |