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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 6352 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾)) | |
2 | isclat.u | . . . . . 6 ⊢ 𝑈 = (lub‘𝐾) | |
3 | 1, 2 | syl6eqr 2812 | . . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈) |
4 | 3 | dmeqd 5481 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈) |
5 | fveq2 6352 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾)) | |
6 | isclat.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 5, 6 | syl6eqr 2812 | . . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵) |
8 | 7 | pweqd 4307 | . . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵) |
9 | 4, 8 | eqeq12d 2775 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵)) |
10 | fveq2 6352 | . . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾)) | |
11 | isclat.g | . . . . . 6 ⊢ 𝐺 = (glb‘𝐾) | |
12 | 10, 11 | syl6eqr 2812 | . . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺) |
13 | 12 | dmeqd 5481 | . . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺) |
14 | 13, 8 | eqeq12d 2775 | . . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵)) |
15 | 9, 14 | anbi12d 749 | . 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 17309 | . 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))} | |
17 | 15, 16 | elrab2 3507 | 1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 𝒫 cpw 4302 dom cdm 5266 ‘cfv 6049 Basecbs 16059 Posetcpo 17141 lubclub 17143 glbcglb 17144 CLatccla 17308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-dm 5276 df-iota 6012 df-fv 6057 df-clat 17309 |
This theorem is referenced by: clatpos 17311 clatlem 17312 clatlubcl2 17314 clatglbcl2 17316 clatl 17317 oduclatb 17345 xrsclat 29989 |
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