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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvresioo | Structured version Visualization version GIF version |
Description: Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvresioo | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℝ D (𝐹 ↾ (𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ (𝐵(,)𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-resscn 10194 | . . . 4 ⊢ ℝ ⊆ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → ℝ ⊆ ℂ) |
3 | simpr 471 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → 𝐹:𝐴⟶ℂ) | |
4 | simpl 468 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → 𝐴 ⊆ ℝ) | |
5 | ioossre 12439 | . . . 4 ⊢ (𝐵(,)𝐶) ⊆ ℝ | |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → (𝐵(,)𝐶) ⊆ ℝ) |
7 | eqid 2770 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
8 | 7 | tgioo2 22825 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
9 | 7, 8 | dvres 23894 | . . 3 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ ℝ ∧ (𝐵(,)𝐶) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐵(,)𝐶)))) |
10 | 2, 3, 4, 6, 9 | syl22anc 1476 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℝ D (𝐹 ↾ (𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐵(,)𝐶)))) |
11 | ioontr 40250 | . . 3 ⊢ ((int‘(topGen‘ran (,)))‘(𝐵(,)𝐶)) = (𝐵(,)𝐶) | |
12 | 11 | reseq2i 5531 | . 2 ⊢ ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ (𝐵(,)𝐶)) |
13 | 10, 12 | syl6eq 2820 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℝ D (𝐹 ↾ (𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ (𝐵(,)𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1630 ⊆ wss 3721 ran crn 5250 ↾ cres 5251 ⟶wf 6027 ‘cfv 6031 (class class class)co 6792 ℂcc 10135 ℝcr 10136 (,)cioo 12379 TopOpenctopn 16289 topGenctg 16305 ℂfldccnfld 19960 intcnt 21041 D cdv 23846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fi 8472 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-fz 12533 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-plusg 16161 df-mulr 16162 df-starv 16163 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-rest 16290 df-topn 16291 df-topgen 16311 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-cnp 21252 df-xms 22344 df-ms 22345 df-limc 23849 df-dv 23850 |
This theorem is referenced by: fouriersw 40959 |
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