dvmptresicc - Metamath Proof Explorer

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Theorem dvmptresicc 25856

Description: Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dvmptresicc	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))

Distinct variable groups: 𝑥,𝐶 𝑥,𝐷 𝜑,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐹(𝑥)

**Proof of Theorem dvmptresicc**
Step	Hyp	Ref	Expression
1		dvmptresicc.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝐴)
2	1	reseq1i 5960	. . . 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 13441	. . . . . 6 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
6		ax-resscn 11179	. . . . . . 7 ⊢ ℝ ⊆ ℂ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
8	5, 7	sstrd 3967	. . . . 5 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
9	8	resmptd 6025	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))
10	2, 9	eqtrid 2781	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))
11	10	oveq2d 7416	. 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)))
12	5	resabs1d 5993	. . . . 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 7416	. . 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 7101	. . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
17	16, 7	fssresd 6742	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ℝ):ℝ⟶ℂ)
18		ssidd 3980	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ)
19		eqid 2734	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20		tgioo4 24731	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21	19, 20	dvres 25851	. . . 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 838	. . 3 ⊢ (𝜑 → (ℝ D ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) = ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
23		reelprrecn 11214	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
25		ssidd 3980	. . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ)
26		dvmptresicc.fdv	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵))
27	26	dmeqd 5883	. . . . . . . 8 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom (𝑥 ∈ ℂ ↦ 𝐵))
28		dvmptresicc.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
29	28	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℂ 𝐵 ∈ ℂ)
30		dmmptg 6229	. . . . . . . . 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 2770	. . . . . . 7 ⊢ (𝜑 → ℂ = dom (ℂ D 𝐹))
33	7, 32	sseqtrd 3993	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹))
34		dvres3 25853	. . . . . 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 838	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
36		iccntr 24748	. . . . . 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 5965	. . . 4 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)))
39		ioossre 13415	. . . . 5 ⊢ (𝐶(,)𝐷) ⊆ ℝ
40		resabs1 5991	. . . . 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 5963	. . . . 5 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)))
43		ioosscn 13416	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℂ
44		resmpt 6022	. . . . . 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 2769	. . . 4 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
47	38, 41, 46	3eqtrd 2773	. . 3 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
48	14, 22, 47	3eqtrd 2773	. 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
49	11, 48	eqtr3d 2771	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 1539 ∈ wcel 2107 ∀wral 3050 ⊆ wss 3924 {cpr 4601 ↦ cmpt 5199 dom cdm 5652 ran crn 5653 ↾ cres 5654 ⟶wf 6524 ‘cfv 6528 (class class class)co 7400 ℂcc 11120 ℝcr 11121 (,)cioo 13354 [,]cicc 13357 TopOpenctopn 17422 topGenctg 17438 ℂ_fldccnfld 21302 intcnt 22942 D cdv 25803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199 ax-pre-sup 11200

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-tp 4604 df-op 4606 df-uni 4882 df-int 4921 df-iun 4967 df-iin 4968 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7857 df-1st 7983 df-2nd 7984 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-1o 8475 df-er 8714 df-map 8837 df-pm 8838 df-en 8955 df-dom 8956 df-sdom 8957 df-fin 8958 df-fi 9418 df-sup 9449 df-inf 9450 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-div 11888 df-nn 12234 df-2 12296 df-3 12297 df-4 12298 df-5 12299 df-6 12300 df-7 12301 df-8 12302 df-9 12303 df-n0 12495 df-z 12582 df-dec 12702 df-uz 12846 df-q 12958 df-rp 13002 df-xneg 13121 df-xadd 13122 df-xmul 13123 df-ioo 13358 df-ico 13360 df-icc 13361 df-fz 13515 df-seq 14010 df-exp 14070 df-cj 15107 df-re 15108 df-im 15109 df-sqrt 15243 df-abs 15244 df-struct 17153 df-slot 17188 df-ndx 17200 df-base 17216 df-plusg 17271 df-mulr 17272 df-starv 17273 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-rest 17423 df-topn 17424 df-topgen 17444 df-psmet 21294 df-xmet 21295 df-met 21296 df-bl 21297 df-mopn 21298 df-fbas 21299 df-fg 21300 df-cnfld 21303 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-lp 23061 df-perf 23062 df-cnp 23153 df-haus 23240 df-fil 23771 df-fm 23863 df-flim 23864 df-flf 23865 df-xms 24246 df-ms 24247 df-limc 25806 df-dv 25807

This theorem is referenced by: resdvopclptsd 41970 itgsincmulx 45939

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