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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 6055 | . . . . . . 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 2768 | . . . . 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 2838 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ) |
| 12 | rescncf 24923 | . . . . . 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 2837 | . . . 4 ⊢ 𝐺 ∈ (𝐴–cn→ℂ) |
| 15 | cncfcdm 24924 | . . . 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 2737 | . . . 4 ⊢ ((abs ∘ − ) ↾ (𝐴 × 𝐴)) = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) | |
| 19 | eqid 2737 | . . . 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 24935 | . . 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 2839 | 1 ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ↦ cmpt 5225 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 − cmin 11492 abscabs 15273 MetOpencmopn 21354 Cn ccn 23232 –cn→ccncf 24902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-cnp 23236 df-cncf 24904 |
| This theorem is referenced by: (None) |
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