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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 6878 | . . 3 ⊢ 𝐺:𝐴⟶𝐵 |
4 | cncfres.2 | . . . 4 ⊢ 𝐵 ⊆ ℂ | |
5 | cncfres.1 | . . . . . . 7 ⊢ 𝐴 ⊆ ℂ | |
6 | resmpt 5907 | . . . . . . 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 2849 | . . . . 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 2912 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ) |
12 | rescncf 23507 | . . . . . 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 2911 | . . . 4 ⊢ 𝐺 ∈ (𝐴–cn→ℂ) |
15 | cncffvrn 23508 | . . . 4 ⊢ ((𝐵 ⊆ ℂ ∧ 𝐺 ∈ (𝐴–cn→ℂ)) → (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵)) | |
16 | 4, 14, 15 | mp2an 690 | . . 3 ⊢ (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵) |
17 | 3, 16 | mpbir 233 | . 2 ⊢ 𝐺 ∈ (𝐴–cn→𝐵) |
18 | eqid 2823 | . . . 4 ⊢ ((abs ∘ − ) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) | |
19 | eqid 2823 | . . . 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 23518 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾)) |
23 | 5, 4, 22 | mp2an 690 | . 2 ⊢ (𝐴–cn→𝐵) = (𝐽 Cn 𝐾) |
24 | 17, 23 | eleqtri 2913 | 1 ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ↦ cmpt 5148 × cxp 5555 ↾ cres 5559 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 − cmin 10872 abscabs 14595 MetOpencmopn 20537 Cn ccn 21834 –cn→ccncf 23486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-cn 21837 df-cnp 21838 df-cncf 23488 |
This theorem is referenced by: (None) |
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