Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cncfcn1cntop | GIF version |
Description: Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.) |
Ref | Expression |
---|---|
cncfcn1.1 | ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
Ref | Expression |
---|---|
cncfcn1cntop | ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3117 | . 2 ⊢ ℂ ⊆ ℂ | |
2 | cncfcn1.1 | . . 3 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) | |
3 | 2 | cntoptopon 12711 | . . . 4 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
4 | 3 | toponrestid 12198 | . . 3 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
5 | 2, 4, 4 | cncfcncntop 12759 | . 2 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)) |
6 | 1, 1, 5 | mp2an 422 | 1 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3071 ∘ ccom 4543 ‘cfv 5123 (class class class)co 5774 ℂcc 7625 − cmin 7940 abscabs 10776 MetOpencmopn 12164 Cn ccn 12364 –cn→ccncf 12736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 ax-arch 7746 ax-caucvg 7747 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 df-inn 8728 df-2 8786 df-3 8787 df-4 8788 df-n0 8985 df-z 9062 df-uz 9334 df-q 9419 df-rp 9449 df-xneg 9566 df-xadd 9567 df-seqfrec 10226 df-exp 10300 df-cj 10621 df-re 10622 df-im 10623 df-rsqrt 10777 df-abs 10778 df-rest 12132 df-topgen 12151 df-psmet 12166 df-xmet 12167 df-met 12168 df-bl 12169 df-mopn 12170 df-top 12175 df-topon 12188 df-bases 12220 df-cn 12367 df-cnp 12368 df-cncf 12737 |
This theorem is referenced by: dvcjbr 12851 |
Copyright terms: Public domain | W3C validator |