Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfres | Structured version Visualization version GIF version |
Description: A continuous function on complex numbers restricted to a subset. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
cncfres.1 | ⊢ 𝐴 ⊆ ℂ |
cncfres.2 | ⊢ 𝐵 ⊆ ℂ |
cncfres.3 | ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐶) |
cncfres.4 | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
cncfres.5 | ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
cncfres.6 | ⊢ 𝐹 ∈ (ℂ–cn→ℂ) |
cncfres.7 | ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) |
cncfres.8 | ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
cncfres | ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfres.4 | . . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
2 | cncfres.5 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) | |
3 | 1, 2 | fmpti 6383 | . . 3 ⊢ 𝐺:𝐴⟶𝐵 |
4 | cncfres.2 | . . . 4 ⊢ 𝐵 ⊆ ℂ | |
5 | cncfres.1 | . . . . . . 7 ⊢ 𝐴 ⊆ ℂ | |
6 | resmpt 5449 | . . . . . . 7 ⊢ (𝐴 ⊆ ℂ → ((𝑥 ∈ ℂ ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
8 | 1, 7 | eqtr4i 2647 | . . . . 5 ⊢ 𝐺 = ((𝑥 ∈ ℂ ↦ 𝐶) ↾ 𝐴) |
9 | cncfres.3 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐶) | |
10 | cncfres.6 | . . . . . . 7 ⊢ 𝐹 ∈ (ℂ–cn→ℂ) | |
11 | 9, 10 | eqeltrri 2698 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ) |
12 | rescncf 22700 | . . . . . 6 ⊢ (𝐴 ⊆ ℂ → ((𝑥 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ) → ((𝑥 ∈ ℂ ↦ 𝐶) ↾ 𝐴) ∈ (𝐴–cn→ℂ))) | |
13 | 5, 11, 12 | mp2 9 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ↦ 𝐶) ↾ 𝐴) ∈ (𝐴–cn→ℂ) |
14 | 8, 13 | eqeltri 2697 | . . . 4 ⊢ 𝐺 ∈ (𝐴–cn→ℂ) |
15 | cncffvrn 22701 | . . . 4 ⊢ ((𝐵 ⊆ ℂ ∧ 𝐺 ∈ (𝐴–cn→ℂ)) → (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵)) | |
16 | 4, 14, 15 | mp2an 708 | . . 3 ⊢ (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵) |
17 | 3, 16 | mpbir 221 | . 2 ⊢ 𝐺 ∈ (𝐴–cn→𝐵) |
18 | eqid 2622 | . . . 4 ⊢ ((abs ∘ − ) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) | |
19 | eqid 2622 | . . . 4 ⊢ ((abs ∘ − ) ↾ (𝐵 × 𝐵)) = ((abs ∘ − ) ↾ (𝐵 × 𝐵)) | |
20 | cncfres.7 | . . . 4 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) | |
21 | cncfres.8 | . . . 4 ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))) | |
22 | 18, 19, 20, 21 | cncfmet 22711 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾)) |
23 | 5, 4, 22 | mp2an 708 | . 2 ⊢ (𝐴–cn→𝐵) = (𝐽 Cn 𝐾) |
24 | 17, 23 | eleqtri 2699 | 1 ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 − cmin 10266 abscabs 13974 MetOpencmopn 19736 Cn ccn 21028 –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 ax-pre-sup 10014 |
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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 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-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-cnp 21032 df-cncf 22681 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |