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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 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
cncfcn.3 | ⊢ 𝐾 = (𝐽 ↾t 𝐴) |
cncfcn.4 | ⊢ 𝐿 = (𝐽 ↾t 𝐵) |
Ref | Expression |
---|---|
cncfcn | ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2793 | . . 3 ⊢ ((abs ∘ − ) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) | |
2 | eqid 2793 | . . 3 ⊢ ((abs ∘ − ) ↾ (𝐵 × 𝐵)) = ((abs ∘ − ) ↾ (𝐵 × 𝐵)) | |
3 | eqid 2793 | . . 3 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) | |
4 | eqid 2793 | . . 3 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))) = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))) | |
5 | 1, 2, 3, 4 | cncfmet 23187 | . 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = ((MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) Cn (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))))) |
6 | cncfcn.3 | . . . 4 ⊢ 𝐾 = (𝐽 ↾t 𝐴) | |
7 | cnxmet 23052 | . . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
8 | simpl 483 | . . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐴 ⊆ ℂ) | |
9 | cncfcn.2 | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
10 | 9 | cnfldtopn 23061 | . . . . . 6 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
11 | 1, 10, 3 | metrest 22805 | . . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ⊆ ℂ) → (𝐽 ↾t 𝐴) = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))) |
12 | 7, 8, 11 | sylancr 587 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐽 ↾t 𝐴) = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))) |
13 | 6, 12 | syl5eq 2841 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))) |
14 | cncfcn.4 | . . . 4 ⊢ 𝐿 = (𝐽 ↾t 𝐵) | |
15 | simpr 485 | . . . . 5 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐵 ⊆ ℂ) | |
16 | 2, 10, 4 | metrest 22805 | . . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ⊆ ℂ) → (𝐽 ↾t 𝐵) = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵)))) |
17 | 7, 15, 16 | sylancr 587 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐽 ↾t 𝐵) = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵)))) |
18 | 14, 17 | syl5eq 2841 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐿 = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵)))) |
19 | 13, 18 | oveq12d 7025 | . 2 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐾 Cn 𝐿) = ((MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) Cn (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))))) |
20 | 5, 19 | eqtr4d 2832 | 1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ⊆ wss 3854 × cxp 5433 ↾ cres 5437 ∘ ccom 5439 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 − cmin 10706 abscabs 14415 ↾t crest 16511 TopOpenctopn 16512 ∞Metcxmet 20200 MetOpencmopn 20205 ℂfldccnfld 20215 Cn ccn 21504 –cn→ccncf 23155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-inf 8743 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-fz 12732 df-seq 13208 df-exp 13268 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-plusg 16395 df-mulr 16396 df-starv 16397 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-rest 16513 df-topn 16514 df-topgen 16534 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-cnfld 20216 df-top 21174 df-topon 21191 df-bases 21226 df-cn 21507 df-cnp 21508 df-cncf 23157 |
This theorem is referenced by: cncfcn1 23189 cncfmptc 23190 cncfmptid 23191 cncfmpt2f 23193 cdivcncf 23196 abscncfALT 23199 cncfcnvcn 23200 cnrehmeo 23228 cncombf 23930 cnmbf 23931 cnlimc 24157 dvcn 24189 dvcnvrelem2 24286 dvcnvre 24287 ftc1cn 24311 psercn 24685 abelth 24700 logcn 24899 dvloglem 24900 efopnlem2 24909 cxpcn 24995 resqrtcn 24999 sqrtcn 25000 loglesqrt 25008 ftalem3 25322 cxpcncf1 31439 ivthALT 33237 knoppcnlem10 33394 knoppcnlem11 33395 ftc1cnnc 34443 areacirclem2 34460 areacirclem4 34462 fsumcncf 41656 ioccncflimc 41663 cncfuni 41664 icocncflimc 41667 cncfdmsn 41668 cncfiooicclem1 41671 cncfiooicc 41672 cxpcncf2 41678 itgsubsticclem 41755 dirkercncflem2 41885 dirkercncflem4 41887 dirkercncf 41888 fourierdlem32 41920 fourierdlem33 41921 fourierdlem62 41949 fourierdlem93 41980 fourierdlem101 41988 fouriercn 42013 |
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