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Mirrors > Home > MPE Home > Th. List > cncfcn | Structured version Visualization version GIF version |
Description: Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
cncfcn.2 | β’ π½ = (TopOpenββfld) |
cncfcn.3 | β’ πΎ = (π½ βΎt π΄) |
cncfcn.4 | β’ πΏ = (π½ βΎt π΅) |
Ref | Expression |
---|---|
cncfcn | β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = (πΎ Cn πΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ ((abs β β ) βΎ (π΄ Γ π΄)) = ((abs β β ) βΎ (π΄ Γ π΄)) | |
2 | eqid 2733 | . . 3 β’ ((abs β β ) βΎ (π΅ Γ π΅)) = ((abs β β ) βΎ (π΅ Γ π΅)) | |
3 | eqid 2733 | . . 3 β’ (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) | |
4 | eqid 2733 | . . 3 β’ (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))) = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))) | |
5 | 1, 2, 3, 4 | cncfmet 24295 | . 2 β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = ((MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) Cn (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))))) |
6 | cncfcn.3 | . . . 4 β’ πΎ = (π½ βΎt π΄) | |
7 | cnxmet 24159 | . . . . 5 β’ (abs β β ) β (βMetββ) | |
8 | simpl 484 | . . . . 5 β’ ((π΄ β β β§ π΅ β β) β π΄ β β) | |
9 | cncfcn.2 | . . . . . . 7 β’ π½ = (TopOpenββfld) | |
10 | 9 | cnfldtopn 24168 | . . . . . 6 β’ π½ = (MetOpenβ(abs β β )) |
11 | 1, 10, 3 | metrest 23903 | . . . . 5 β’ (((abs β β ) β (βMetββ) β§ π΄ β β) β (π½ βΎt π΄) = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄)))) |
12 | 7, 8, 11 | sylancr 588 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π½ βΎt π΄) = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄)))) |
13 | 6, 12 | eqtrid 2785 | . . 3 β’ ((π΄ β β β§ π΅ β β) β πΎ = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄)))) |
14 | cncfcn.4 | . . . 4 β’ πΏ = (π½ βΎt π΅) | |
15 | simpr 486 | . . . . 5 β’ ((π΄ β β β§ π΅ β β) β π΅ β β) | |
16 | 2, 10, 4 | metrest 23903 | . . . . 5 β’ (((abs β β ) β (βMetββ) β§ π΅ β β) β (π½ βΎt π΅) = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅)))) |
17 | 7, 15, 16 | sylancr 588 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π½ βΎt π΅) = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅)))) |
18 | 14, 17 | eqtrid 2785 | . . 3 β’ ((π΄ β β β§ π΅ β β) β πΏ = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅)))) |
19 | 13, 18 | oveq12d 7379 | . 2 β’ ((π΄ β β β§ π΅ β β) β (πΎ Cn πΏ) = ((MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) Cn (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))))) |
20 | 5, 19 | eqtr4d 2776 | 1 β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = (πΎ Cn πΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3914 Γ cxp 5635 βΎ cres 5639 β ccom 5641 βcfv 6500 (class class class)co 7361 βcc 11057 β cmin 11393 abscabs 15128 βΎt crest 17310 TopOpenctopn 17311 βMetcxmet 20804 MetOpencmopn 20809 βfldccnfld 20819 Cn ccn 22598 βcnβccncf 24262 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-rest 17312 df-topn 17313 df-topgen 17333 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-cnfld 20820 df-top 22266 df-topon 22283 df-bases 22319 df-cn 22601 df-cnp 22602 df-cncf 24264 |
This theorem is referenced by: cncfcn1 24297 cncfmptc 24298 cncfmptid 24299 cncfmpt2f 24301 cdivcncf 24307 abscncfALT 24310 cncfcnvcn 24311 cnrehmeo 24339 cncombf 25045 cnmbf 25046 cnlimc 25275 dvcn 25308 dvcnvrelem2 25405 dvcnvre 25406 ftc1cn 25430 psercn 25808 abelth 25823 logcn 26025 dvloglem 26026 efopnlem2 26035 cxpcn 26121 resqrtcn 26125 sqrtcn 26126 loglesqrt 26134 ftalem3 26447 cxpcncf1 33272 ivthALT 34860 knoppcnlem10 35018 knoppcnlem11 35019 ftc1cnnc 36200 areacirclem2 36217 areacirclem4 36219 fsumcncf 44209 ioccncflimc 44216 cncfuni 44217 icocncflimc 44220 cncfdmsn 44221 cncfiooicclem1 44224 cncfiooicc 44225 cxpcncf2 44230 itgsubsticclem 44306 dirkercncflem2 44435 dirkercncflem4 44437 dirkercncf 44438 fourierdlem32 44470 fourierdlem33 44471 fourierdlem62 44499 fourierdlem93 44530 fourierdlem101 44538 fouriercn 44563 |
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