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Mirrors > Home > MPE Home > Th. List > cncfcn1 | Structured version Visualization version GIF version |
Description: Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) |
Ref | Expression |
---|---|
cncfcn1.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
cncfcn1 | ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3999 | . 2 ⊢ ℂ ⊆ ℂ | |
2 | cncfcn1.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 24748 | . . . 4 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
4 | 3 | toponrestid 22872 | . . 3 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
5 | 2, 4, 4 | cncfcn 24879 | . 2 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)) |
6 | 1, 1, 5 | mp2an 690 | 1 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊆ wss 3944 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 TopOpenctopn 17411 ℂfldccnfld 21301 Cn ccn 23177 –cn→ccncf 24845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-seq 14008 df-exp 14068 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17189 df-plusg 17254 df-mulr 17255 df-starv 17256 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-rest 17412 df-topn 17413 df-topgen 17433 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-cnfld 21302 df-top 22845 df-topon 22862 df-topsp 22884 df-bases 22898 df-cn 23180 df-cnp 23181 df-xms 24275 df-ms 24276 df-cncf 24847 |
This theorem is referenced by: expcncf 24896 dvcjbr 25930 plycn 26245 plycnOLD 26246 psercn2 26409 psercn2OLD 26410 efopn 26642 cxpcn 26729 cxpcnOLD 26730 efrlim 26951 efrlimOLD 26952 pntlem3 27592 cvxpconn 34985 sinccvglem 35409 dvtanlem 37275 climexp 45133 fprodsub2cncf 45433 fprodadd2cncf 45434 |
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