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Mathbox for Jeff Madsen |
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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 7132 | . . 3 ⊢ 𝐺:𝐴⟶𝐵 |
4 | cncfres.2 | . . . 4 ⊢ 𝐵 ⊆ ℂ | |
5 | cncfres.1 | . . . . . . 7 ⊢ 𝐴 ⊆ ℂ | |
6 | resmpt 6057 | . . . . . . 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 2766 | . . . . 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 2836 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ) |
12 | rescncf 24937 | . . . . . 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 2835 | . . . 4 ⊢ 𝐺 ∈ (𝐴–cn→ℂ) |
15 | cncfcdm 24938 | . . . 4 ⊢ ((𝐵 ⊆ ℂ ∧ 𝐺 ∈ (𝐴–cn→ℂ)) → (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵)) | |
16 | 4, 14, 15 | mp2an 692 | . . 3 ⊢ (𝐺 ∈ (𝐴–cn→𝐵) ↔ 𝐺:𝐴⟶𝐵) |
17 | 3, 16 | mpbir 231 | . 2 ⊢ 𝐺 ∈ (𝐴–cn→𝐵) |
18 | eqid 2735 | . . . 4 ⊢ ((abs ∘ − ) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) | |
19 | eqid 2735 | . . . 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 24949 | . . 3 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾)) |
23 | 5, 4, 22 | mp2an 692 | . 2 ⊢ (𝐴–cn→𝐵) = (𝐽 Cn 𝐾) |
24 | 17, 23 | eleqtri 2837 | 1 ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ↦ cmpt 5231 × cxp 5687 ↾ cres 5691 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 − cmin 11490 abscabs 15270 MetOpencmopn 21372 Cn ccn 23248 –cn→ccncf 24916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cn 23251 df-cnp 23252 df-cncf 24918 |
This theorem is referenced by: (None) |
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