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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 25916 | . . 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 | elmpocl 7674 | . 2 ⊢ (𝐶 ∈ (𝐹 limℂ 𝐵) → (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ)) |
3 | cnex 11234 | . . . . 5 ⊢ ℂ ∈ V | |
4 | 3, 3 | elpm2 8913 | . . . 4 ⊢ (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)) |
5 | 4 | anbi1i 624 | . . 3 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ 𝐵 ∈ ℂ)) |
6 | df-3an 1088 | . . 3 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ 𝐵 ∈ ℂ)) | |
7 | 5, 6 | bitr4i 278 | . 2 ⊢ ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
8 | 2, 7 | sylib 218 | 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 395 ∧ w3a 1086 ∈ wcel 2106 {cab 2712 [wsbc 3791 ∪ cun 3961 ⊆ wss 3963 ifcif 4531 {csn 4631 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑pm cpm 8866 ℂcc 11151 ↾t crest 17467 TopOpenctopn 17468 ℂfldccnfld 21382 CnP ccnp 23249 limℂ climc 25912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8868 df-limc 25916 |
This theorem is referenced by: limccl 25925 limcdif 25926 limcresi 25935 limcres 25936 limccnp 25941 limccnp2 25942 limcco 25943 limcun 25945 mullimc 45572 limccog 45576 mullimcf 45579 limcperiod 45584 limcmptdm 45591 neglimc 45603 addlimc 45604 0ellimcdiv 45605 reclimc 45609 |
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