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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 5995 | . . . 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 13470 | . . . . . 6 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ) |
6 | ax-resscn 11209 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | sstrd 4005 | . . . . 5 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ) |
9 | 8 | resmptd 6059 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) |
10 | 2, 9 | eqtrid 2786 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) |
11 | 10 | oveq2d 7446 | . 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))) |
12 | 5 | resabs1d 6027 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)) = (𝐹 ↾ (𝐶[,]𝐷))) |
13 | 12 | eqcomd 2740 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) |
14 | 13 | oveq2d 7446 | . . 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 7133 | . . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
17 | 16, 7 | fssresd 6775 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ ℝ):ℝ⟶ℂ) |
18 | ssidd 4018 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
19 | eqid 2734 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
20 | 19 | tgioo2 24838 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
21 | 19, 20 | dvres 25960 | . . . 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 839 | . . 3 ⊢ (𝜑 → (ℝ D ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) = ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)))) |
23 | reelprrecn 11244 | . . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ} | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
25 | ssidd 4018 | . . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
26 | dvmptresicc.fdv | . . . . . . . . 9 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵)) | |
27 | 26 | dmeqd 5918 | . . . . . . . 8 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom (𝑥 ∈ ℂ ↦ 𝐵)) |
28 | dvmptresicc.b | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) | |
29 | 28 | ralrimiva 3143 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℂ 𝐵 ∈ ℂ) |
30 | dmmptg 6263 | . . . . . . . . 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 2775 | . . . . . . 7 ⊢ (𝜑 → ℂ = dom (ℂ D 𝐹)) |
33 | 7, 32 | sseqtrd 4035 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹)) |
34 | dvres3 25962 | . . . . . 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 839 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ)) |
36 | iccntr 24856 | . . . . . 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 6000 | . . . 4 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷))) |
39 | ioossre 13444 | . . . . 5 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
40 | resabs1 6026 | . . . . 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 5998 | . . . . 5 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷))) |
43 | ioosscn 13445 | . . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℂ | |
44 | resmpt 6056 | . . . . . 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 2774 | . . . 4 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
47 | 38, 41, 46 | 3eqtrd 2778 | . . 3 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
48 | 14, 22, 47 | 3eqtrd 2778 | . 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
49 | 11, 48 | eqtr3d 2776 | 1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ⊆ wss 3962 {cpr 4632 ↦ cmpt 5230 dom cdm 5688 ran crn 5689 ↾ cres 5690 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 (,)cioo 13383 [,]cicc 13386 TopOpenctopn 17467 topGenctg 17483 ℂfldccnfld 21381 intcnt 23040 D cdv 25912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cnp 23251 df-haus 23338 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-limc 25915 df-dv 25916 |
This theorem is referenced by: resdvopclptsd 42009 itgsincmulx 45929 |
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