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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 5814 | . . . 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 12812 | . . . . . 6 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ) |
6 | ax-resscn 10583 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | sstrd 3925 | . . . . 5 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ) |
9 | 8 | resmptd 5875 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) |
10 | 2, 9 | syl5eq 2845 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) |
11 | 10 | oveq2d 7151 | . 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))) |
12 | 5 | resabs1d 5849 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)) = (𝐹 ↾ (𝐶[,]𝐷))) |
13 | 12 | eqcomd 2804 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) |
14 | 13 | oveq2d 7151 | . . 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 6855 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
17 | 16, 7 | fssresd 6519 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ ℝ):ℝ⟶ℂ) |
18 | ssidd 3938 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
19 | eqid 2798 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
20 | 19 | tgioo2 23408 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
21 | 19, 20 | dvres 24514 | . . . 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 10618 | . . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ} | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
25 | ssidd 3938 | . . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
26 | dvmptresicc.fdv | . . . . . . . . 9 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵)) | |
27 | 26 | dmeqd 5738 | . . . . . . . 8 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom (𝑥 ∈ ℂ ↦ 𝐵)) |
28 | dvmptresicc.b | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) | |
29 | 28 | ralrimiva 3149 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℂ 𝐵 ∈ ℂ) |
30 | dmmptg 6063 | . . . . . . . . 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 2834 | . . . . . . 7 ⊢ (𝜑 → ℂ = dom (ℂ D 𝐹)) |
33 | 7, 32 | sseqtrd 3955 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹)) |
34 | dvres3 24516 | . . . . . 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 23426 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷)) | |
37 | 3, 4, 36 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷)) |
38 | 35, 37 | reseq12d 5819 | . . . 4 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷))) |
39 | ioossre 12786 | . . . . 5 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
40 | resabs1 5848 | . . . . 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 5817 | . . . . 5 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷))) |
43 | ioosscn 12787 | . . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℂ | |
44 | resmpt 5872 | . . . . . 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 2833 | . . . 4 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
47 | 38, 41, 46 | 3eqtrd 2837 | . . 3 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
48 | 14, 22, 47 | 3eqtrd 2837 | . 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
49 | 11, 48 | eqtr3d 2835 | 1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 {cpr 4527 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 (,)cioo 12726 [,]cicc 12729 TopOpenctopn 16687 topGenctg 16703 ℂfldccnfld 20091 intcnt 21622 D cdv 24466 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cnp 21833 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-limc 24469 df-dv 24470 |
This theorem is referenced by: resdvopclptsd 39316 itgsincmulx 42616 |
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