![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > limcrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
limcrcl | ⊢ (𝐶 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-limc 23849 | . . 3 ⊢ limℂ = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∣ [(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓‘𝑧))) ∈ (((𝑗 ↾t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)}) | |
2 | 1 | elmpt2cl 7042 | . 2 ⊢ (𝐶 ∈ (𝐹 limℂ 𝐵) → (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ)) |
3 | cnex 10229 | . . . . 5 ⊢ ℂ ∈ V | |
4 | 3, 3 | elpm2 8057 | . . . 4 ⊢ (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)) |
5 | 4 | anbi1i 733 | . . 3 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ 𝐵 ∈ ℂ)) |
6 | df-3an 1074 | . . 3 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ 𝐵 ∈ ℂ)) | |
7 | 5, 6 | bitr4i 267 | . 2 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
8 | 2, 7 | sylib 208 | 1 ⊢ (𝐶 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2139 {cab 2746 [wsbc 3576 ∪ cun 3713 ⊆ wss 3715 ifcif 4230 {csn 4321 ↦ cmpt 4881 dom cdm 5266 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 ↑pm cpm 8026 ℂcc 10146 ↾t crest 16303 TopOpenctopn 16304 ℂfldccnfld 19968 CnP ccnp 21251 limℂ climc 23845 |
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-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 |
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-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 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-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-pm 8028 df-limc 23849 |
This theorem is referenced by: limccl 23858 limcdif 23859 limcresi 23868 limcres 23869 limccnp 23874 limccnp2 23875 limcco 23876 limcun 23878 mullimc 40369 limccog 40373 mullimcf 40376 limcperiod 40381 limcmptdm 40388 neglimc 40400 addlimc 40401 0ellimcdiv 40402 reclimc 40406 |
Copyright terms: Public domain | W3C validator |