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Mirrors > Home > MPE Home > Th. List > cncffvrn | Structured version Visualization version GIF version |
Description: Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
cncffvrn | ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ 𝐹:𝐴⟶𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfrss 22694 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) | |
2 | 1 | adantl 482 | . . 3 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐴 ⊆ ℂ) |
3 | simpl 473 | . . 3 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → 𝐶 ⊆ ℂ) | |
4 | elcncf2 22693 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐶 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) | |
5 | 2, 3, 4 | syl2anc 693 | . 2 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) |
6 | cncfi 22697 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)) | |
7 | 6 | 3expb 1266 | . . . . 5 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+)) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)) |
8 | 7 | ralrimivva 2971 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)) |
9 | 8 | adantl 482 | . . 3 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)) |
10 | 9 | biantrud 528 | . 2 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹:𝐴⟶𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) |
11 | 5, 10 | bitr4d 271 | 1 ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ 𝐹:𝐴⟶𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 < clt 10074 − cmin 10266 ℝ+crp 11832 abscabs 13974 –cn→ccncf 22679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 df-im 13841 df-abs 13976 df-cncf 22681 |
This theorem is referenced by: cncfss 22702 cncfmpt2ss 22718 rolle 23753 dvlipcn 23757 c1lip2 23761 dvivthlem1 23771 dvivth 23773 lhop1lem 23776 dvcnvrelem2 23781 dvfsumlem2 23790 itgsubstlem 23811 efcvx 24203 dvrelog 24383 relogcn 24384 logcn 24393 dvlog 24397 logccv 24409 resqrtcn 24490 loglesqrt 24499 lgamgulmlem2 24756 rpsqrtcn 30671 fdvneggt 30678 fdvnegge 30680 logdivsqrle 30728 knoppcn2 32527 areacirclem4 33503 cncfres 33564 cncfmptssg 40083 resincncf 40088 cncfcompt 40096 cncfiooiccre 40108 dvdivcncf 40142 dvbdfbdioolem1 40143 ioodvbdlimc1lem2 40147 ioodvbdlimc2lem 40149 itgsbtaddcnst 40198 fourierdlem58 40381 fourierdlem59 40382 fourierdlem62 40385 fourierdlem68 40391 fourierdlem76 40399 fourierdlem78 40401 fourierdlem83 40406 fourierdlem101 40424 fourierdlem112 40435 fouriercn 40449 |
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