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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 | β’ π½ = (MetOpenβ(abs β β )) |
Ref | Expression |
---|---|
cncfcn1cntop | β’ (ββcnββ) = (π½ Cn π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3190 | . 2 β’ β β β | |
2 | cncfcn1.1 | . . 3 β’ π½ = (MetOpenβ(abs β β )) | |
3 | 2 | cntoptopon 14436 | . . . 4 β’ π½ β (TopOnββ) |
4 | 3 | toponrestid 13925 | . . 3 β’ π½ = (π½ βΎt β) |
5 | 2, 4, 4 | cncfcncntop 14484 | . 2 β’ ((β β β β§ β β β) β (ββcnββ) = (π½ Cn π½)) |
6 | 1, 1, 5 | mp2an 426 | 1 β’ (ββcnββ) = (π½ Cn π½) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 β wss 3144 β ccom 4645 βcfv 5232 (class class class)co 5892 βcc 7829 β cmin 8148 abscabs 11026 MetOpencmopn 13822 Cn ccn 14089 βcnβccncf 14461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-mulrcl 7930 ax-addcom 7931 ax-mulcom 7932 ax-addass 7933 ax-mulass 7934 ax-distr 7935 ax-i2m1 7936 ax-0lt1 7937 ax-1rid 7938 ax-0id 7939 ax-rnegex 7940 ax-precex 7941 ax-cnre 7942 ax-pre-ltirr 7943 ax-pre-ltwlin 7944 ax-pre-lttrn 7945 ax-pre-apti 7946 ax-pre-ltadd 7947 ax-pre-mulgt0 7948 ax-pre-mulext 7949 ax-arch 7950 ax-caucvg 7951 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-isom 5241 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-map 6669 df-sup 7003 df-inf 7004 df-pnf 8014 df-mnf 8015 df-xr 8016 df-ltxr 8017 df-le 8018 df-sub 8150 df-neg 8151 df-reap 8552 df-ap 8559 df-div 8650 df-inn 8940 df-2 8998 df-3 8999 df-4 9000 df-n0 9197 df-z 9274 df-uz 9549 df-q 9640 df-rp 9674 df-xneg 9792 df-xadd 9793 df-seqfrec 10466 df-exp 10540 df-cj 10871 df-re 10872 df-im 10873 df-rsqrt 11027 df-abs 11028 df-rest 12719 df-topgen 12738 df-psmet 13824 df-xmet 13825 df-met 13826 df-bl 13827 df-mopn 13828 df-top 13902 df-topon 13915 df-bases 13947 df-cn 14092 df-cnp 14093 df-cncf 14462 |
This theorem is referenced by: dvcjbr 14576 |
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