| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > cncff | Structured version Visualization version GIF version | ||
| Description: A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| Ref | Expression |
|---|---|
| cncff | ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cncfrss 24917 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) | |
| 2 | cncfrss2 24918 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
| 3 | elcncf 24915 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))) | |
| 4 | 1, 2, 3 | syl2anc 584 | . . 3 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))) |
| 5 | 4 | ibi 267 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))) |
| 6 | 5 | simpld 494 | 1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 < clt 11295 − cmin 11492 ℝ+crp 13034 abscabs 15273 –cn→ccncf 24902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-cncf 24904 |
| This theorem is referenced by: cncfss 24925 climcncf 24926 cncfco 24933 cncfcompt2 24934 cncfmpt1f 24940 cncfmpt2ss 24942 negfcncf 24950 divcncf 25482 ivth2 25490 ivthicc 25493 evthicc2 25495 cniccbdd 25496 volivth 25642 cncombf 25693 cnmbf 25694 cniccibl 25876 cnicciblnc 25878 cnmptlimc 25925 cpnord 25971 cpnres 25973 dvrec 25993 rollelem 26027 rolle 26028 cmvth 26029 cmvthOLD 26030 mvth 26031 dvlip 26032 c1liplem1 26035 c1lip1 26036 c1lip2 26037 dveq0 26039 dvgt0lem1 26041 dvgt0lem2 26042 dvgt0 26043 dvlt0 26044 dvge0 26045 dvle 26046 dvivthlem1 26047 dvivth 26049 dvne0 26050 dvne0f1 26051 dvcnvrelem1 26056 dvcnvrelem2 26057 dvcnvre 26058 dvcvx 26059 dvfsumle 26060 dvfsumleOLD 26061 dvfsumge 26062 dvfsumabs 26063 ftc1cn 26084 ftc2 26085 ftc2ditglem 26086 ftc2ditg 26087 itgparts 26088 itgsubstlem 26089 itgsubst 26090 ulmcn 26442 psercn 26470 pserdvlem2 26472 pserdv 26473 sincn 26488 coscn 26489 logtayl 26702 dvcncxp1 26785 leibpi 26985 lgamgulmlem2 27073 ftc2re 34613 fdvposlt 34614 fdvneggt 34615 fdvposle 34616 fdvnegge 34617 ivthALT 36336 knoppcld 36506 knoppndv 36535 ftc1cnnclem 37698 ftc1cnnc 37699 ftc2nc 37709 3factsumint 42026 intlewftc 42062 dvle2 42073 cnioobibld 43226 evthiccabs 45509 cncfmptss 45602 mulc1cncfg 45604 expcnfg 45606 mulcncff 45885 cncfshift 45889 subcncff 45895 cncfcompt 45898 addcncff 45899 cncficcgt0 45903 divcncff 45906 cncfiooicclem1 45908 cncfiooiccre 45910 cncfioobd 45912 dvsubcncf 45939 dvmulcncf 45940 dvdivcncf 45942 ioodvbdlimc1lem1 45946 cnbdibl 45977 itgsubsticclem 45990 itgsubsticc 45991 itgioocnicc 45992 iblcncfioo 45993 itgiccshift 45995 itgsbtaddcnst 45997 fourierdlem18 46140 fourierdlem32 46154 fourierdlem33 46155 fourierdlem39 46161 fourierdlem48 46169 fourierdlem49 46170 fourierdlem58 46179 fourierdlem59 46180 fourierdlem71 46192 fourierdlem73 46194 fourierdlem81 46202 fourierdlem84 46205 fourierdlem85 46206 fourierdlem88 46209 fourierdlem94 46215 fourierdlem97 46218 fourierdlem101 46222 fourierdlem103 46224 fourierdlem104 46225 fourierdlem111 46232 fourierdlem112 46233 fourierdlem113 46234 fouriercn 46247 |
| Copyright terms: Public domain | W3C validator |