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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 3942 | . 2 ⊢ ℂ ⊆ ℂ | |
2 | cncfcn1.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 23956 | . . . 4 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
4 | 3 | toponrestid 22080 | . . 3 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
5 | 2, 4, 4 | cncfcn 24083 | . 2 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)) |
6 | 1, 1, 5 | mp2an 689 | 1 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊆ wss 3886 ‘cfv 6426 (class class class)co 7267 ℂcc 10879 TopOpenctopn 17142 ℂfldccnfld 20607 Cn ccn 22385 –cn→ccncf 24049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-sup 9188 df-inf 9189 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-fz 13250 df-seq 13732 df-exp 13793 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-struct 16858 df-slot 16893 df-ndx 16905 df-base 16923 df-plusg 16985 df-mulr 16986 df-starv 16987 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-rest 17143 df-topn 17144 df-topgen 17164 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cn 22388 df-cnp 22389 df-xms 23483 df-ms 23484 df-cncf 24051 |
This theorem is referenced by: expcncf 24099 dvcjbr 25123 plycn 25432 psercn2 25592 efopn 25823 cxpcn 25908 efrlim 26129 pntlem3 26767 sinccvglem 33638 dvtanlem 35834 climexp 43127 fprodsub2cncf 43427 fprodadd2cncf 43428 |
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