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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 6291 | . . 3 ⊢ 𝐺:𝐴⟶𝐵 |
4 | cncfres.2 | . . . 4 ⊢ 𝐵 ⊆ ℂ | |
5 | cncfres.1 | . . . . . . 7 ⊢ 𝐴 ⊆ ℂ | |
6 | resmpt 5369 | . . . . . . 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 2635 | . . . . 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 2685 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ) |
12 | rescncf 22508 | . . . . . 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 2684 | . . . 4 ⊢ 𝐺 ∈ (𝐴–cn→ℂ) |
15 | cncffvrn 22509 | . . . 4 ⊢ ((𝐵 ⊆ ℂ ∧ 𝐺 ∈ (𝐴–cn→ℂ)) → (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵)) | |
16 | 4, 14, 15 | mp2an 704 | . . 3 ⊢ (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵) |
17 | 3, 16 | mpbir 220 | . 2 ⊢ 𝐺 ∈ (𝐴–cn→𝐵) |
18 | eqid 2610 | . . . 4 ⊢ ((abs ∘ − ) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) | |
19 | eqid 2610 | . . . 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 22519 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾)) |
23 | 5, 4, 22 | mp2an 704 | . 2 ⊢ (𝐴–cn→𝐵) = (𝐽 Cn 𝐾) |
24 | 17, 23 | eleqtri 2686 | 1 ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 − cmin 10145 abscabs 13822 MetOpencmopn 19557 Cn ccn 20838 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-cnp 20842 df-cncf 22489 |
This theorem is referenced by: (None) |
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