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Mirrors > Home > MPE Home > Th. List > dvmptresicc | Structured version Visualization version GIF version |
Description: Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvmptresicc.f | ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐴) |
dvmptresicc.a | ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) |
dvmptresicc.fdv | ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵)) |
dvmptresicc.b | ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) |
dvmptresicc.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
dvmptresicc.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
Ref | Expression |
---|---|
dvmptresicc | ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptresicc.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐴) | |
2 | 1 | reseq1i 5975 | . . . 4 ⊢ (𝐹 ↾ (𝐶[,]𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) |
3 | dvmptresicc.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | dvmptresicc.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
5 | 3, 4 | iccssred 13407 | . . . . . 6 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ) |
6 | ax-resscn 11163 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | sstrd 3991 | . . . . 5 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ) |
9 | 8 | resmptd 6038 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) |
10 | 2, 9 | eqtrid 2784 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) |
11 | 10 | oveq2d 7421 | . 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))) |
12 | 5 | resabs1d 6010 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)) = (𝐹 ↾ (𝐶[,]𝐷))) |
13 | 12 | eqcomd 2738 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) |
14 | 13 | oveq2d 7421 | . . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)))) |
15 | dvmptresicc.a | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) | |
16 | 15, 1 | fmptd 7110 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
17 | 16, 7 | fssresd 6755 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ ℝ):ℝ⟶ℂ) |
18 | ssidd 4004 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
19 | eqid 2732 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
20 | 19 | tgioo2 24310 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
21 | 19, 20 | dvres 25419 | . . . 4 ⊢ (((ℝ ⊆ ℂ ∧ (𝐹 ↾ ℝ):ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ)) → (ℝ D ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) = ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)))) |
22 | 7, 17, 18, 5, 21 | syl22anc 837 | . . 3 ⊢ (𝜑 → (ℝ D ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) = ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)))) |
23 | reelprrecn 11198 | . . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ} | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
25 | ssidd 4004 | . . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
26 | dvmptresicc.fdv | . . . . . . . . 9 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵)) | |
27 | 26 | dmeqd 5903 | . . . . . . . 8 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom (𝑥 ∈ ℂ ↦ 𝐵)) |
28 | dvmptresicc.b | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) | |
29 | 28 | ralrimiva 3146 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℂ 𝐵 ∈ ℂ) |
30 | dmmptg 6238 | . . . . . . . . 9 ⊢ (∀𝑥 ∈ ℂ 𝐵 ∈ ℂ → dom (𝑥 ∈ ℂ ↦ 𝐵) = ℂ) | |
31 | 29, 30 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ ℂ ↦ 𝐵) = ℂ) |
32 | 27, 31 | eqtr2d 2773 | . . . . . . 7 ⊢ (𝜑 → ℂ = dom (ℂ D 𝐹)) |
33 | 7, 32 | sseqtrd 4021 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹)) |
34 | dvres3 25421 | . . . . . 6 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ 𝐹:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D 𝐹))) → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ)) | |
35 | 24, 16, 25, 33, 34 | syl22anc 837 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ)) |
36 | iccntr 24328 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷)) | |
37 | 3, 4, 36 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷)) |
38 | 35, 37 | reseq12d 5980 | . . . 4 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷))) |
39 | ioossre 13381 | . . . . 5 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
40 | resabs1 6009 | . . . . 5 ⊢ ((𝐶(,)𝐷) ⊆ ℝ → (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)) = ((ℂ D 𝐹) ↾ (𝐶(,)𝐷))) | |
41 | 39, 40 | mp1i 13 | . . . 4 ⊢ (𝜑 → (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)) = ((ℂ D 𝐹) ↾ (𝐶(,)𝐷))) |
42 | 26 | reseq1d 5978 | . . . . 5 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷))) |
43 | ioosscn 13382 | . . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℂ | |
44 | resmpt 6035 | . . . . . 6 ⊢ ((𝐶(,)𝐷) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) | |
45 | 43, 44 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
46 | 42, 45 | eqtrd 2772 | . . . 4 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
47 | 38, 41, 46 | 3eqtrd 2776 | . . 3 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
48 | 14, 22, 47 | 3eqtrd 2776 | . 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
49 | 11, 48 | eqtr3d 2774 | 1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3947 {cpr 4629 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 (,)cioo 13320 [,]cicc 13323 TopOpenctopn 17363 topGenctg 17379 ℂfldccnfld 20936 intcnt 22512 D cdv 25371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cnp 22723 df-haus 22810 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-limc 25374 df-dv 25375 |
This theorem is referenced by: resdvopclptsd 40881 itgsincmulx 44676 |
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