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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 24725 | . . . . 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 24723 | . . . . 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 499 | . . 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 3888 | . 2 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝑥 ∈ ℂ) |
9 | 8 | ssriv 3891 | 1 ⊢ (𝐹 limℂ 𝐵) ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 {cab 2714 ∪ cun 3851 ⊆ wss 3853 ifcif 4425 {csn 4527 ↦ cmpt 5120 dom cdm 5536 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ↾t crest 16879 TopOpenctopn 16880 ℂfldccnfld 20317 CnP ccnp 22076 limℂ climc 24713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fi 9005 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-fz 13061 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-rest 16881 df-topn 16882 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cnp 22079 df-xms 23172 df-ms 23173 df-limc 24717 |
This theorem is referenced by: ellimc2 24728 limcres 24737 limcco 24744 limciun 24745 limcun 24746 dvfval 24748 dvcl 24750 lhop1lem 24864 mullimc 42775 limcdm0 42777 limccog 42779 mullimcf 42782 limcperiod 42787 limcrecl 42788 limcleqr 42803 neglimc 42806 addlimc 42807 limclner 42810 sublimc 42811 reclimc 42812 divlimc 42815 cncfiooicclem1 43052 cncfiooicc 43053 itgioocnicc 43136 iblcncfioo 43137 fourierdlem60 43325 fourierdlem61 43326 fourierdlem73 43338 fourierdlem74 43339 fourierdlem75 43340 fourierdlem81 43346 fourierdlem103 43368 fourierdlem104 43369 fourierdlem112 43377 |
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