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Mirrors > Home > MPE Home > Th. List > limccl | Structured version Visualization version GIF version |
Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
limccl | ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcrcl 25184 | . . . . 5 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) | |
2 | eqid 2736 | . . . . . 6 ⊢ ((TopOpen‘ℂfld) ↾t (dom 𝐹 ∪ {𝐵})) = ((TopOpen‘ℂfld) ↾t (dom 𝐹 ∪ {𝐵})) | |
3 | eqid 2736 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
4 | 2, 3 | limcfval 25182 | . . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 limℂ 𝐵) = {𝑦 ∣ (𝑧 ∈ (dom 𝐹 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹‘𝑧))) ∈ ((((TopOpen‘ℂfld) ↾t (dom 𝐹 ∪ {𝐵})) CnP (TopOpen‘ℂfld))‘𝐵)} ∧ (𝐹 limℂ 𝐵) ⊆ ℂ)) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → ((𝐹 limℂ 𝐵) = {𝑦 ∣ (𝑧 ∈ (dom 𝐹 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹‘𝑧))) ∈ ((((TopOpen‘ℂfld) ↾t (dom 𝐹 ∪ {𝐵})) CnP (TopOpen‘ℂfld))‘𝐵)} ∧ (𝐹 limℂ 𝐵) ⊆ ℂ)) |
6 | 5 | simprd 496 | . . 3 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹 limℂ 𝐵) ⊆ ℂ) |
7 | id 22 | . . 3 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝑥 ∈ (𝐹 limℂ 𝐵)) | |
8 | 6, 7 | sseldd 3943 | . 2 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝑥 ∈ ℂ) |
9 | 8 | ssriv 3946 | 1 ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cab 2713 ∪ cun 3906 ⊆ wss 3908 ifcif 4484 {csn 4584 ↦ cmpt 5186 dom cdm 5631 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 ↾t crest 17256 TopOpenctopn 17257 ℂfldccnfld 20743 CnP ccnp 22522 limℂ climc 25172 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fi 9305 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-fz 13379 df-seq 13861 df-exp 13922 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-struct 16973 df-slot 17008 df-ndx 17020 df-base 17038 df-plusg 17100 df-mulr 17101 df-starv 17102 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-rest 17258 df-topn 17259 df-topgen 17279 df-psmet 20735 df-xmet 20736 df-met 20737 df-bl 20738 df-mopn 20739 df-cnfld 20744 df-top 22189 df-topon 22206 df-topsp 22228 df-bases 22242 df-cnp 22525 df-xms 23619 df-ms 23620 df-limc 25176 |
This theorem is referenced by: ellimc2 25187 limcres 25196 limcco 25203 limciun 25204 limcun 25205 dvfval 25207 dvcl 25209 lhop1lem 25323 mullimc 43752 limcdm0 43754 limccog 43756 mullimcf 43759 limcperiod 43764 limcrecl 43765 limcleqr 43780 neglimc 43783 addlimc 43784 limclner 43787 sublimc 43788 reclimc 43789 divlimc 43792 cncfiooicclem1 44029 cncfiooicc 44030 itgioocnicc 44113 iblcncfioo 44114 fourierdlem60 44302 fourierdlem61 44303 fourierdlem73 44315 fourierdlem74 44316 fourierdlem75 44317 fourierdlem81 44323 fourierdlem103 44345 fourierdlem104 44346 fourierdlem112 44354 |
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