dvmptresicc - Metamath Proof Explorer

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Theorem dvmptresicc 26144

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 5968	. . . 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 13488	. . . . . 6 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
6		ax-resscn 11182	. . . . . . 7 ⊢ ℝ ⊆ ℂ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
8	5, 7	sstrd 3941	. . . . 5 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
9	8	resmptd 6036	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))
10	2, 9	eqtrid 2807	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))
11	10	oveq2d 7430	. 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)))
12	5	resabs1d 6001	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)) = (𝐹 ↾ (𝐶[,]𝐷)))
13	12	eqcomd 2766	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)))
14	13	oveq2d 7430	. . 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 7108	. . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
17	16, 7	fssresd 6743	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ℝ):ℝ⟶ℂ)
18		ssidd 3954	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ)
19		eqid 2760	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20		tgioo4 25032	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21	19, 20	dvres 26139	. . . 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 852	. . 3 ⊢ (𝜑 → (ℝ D ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷))) = ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))))
23		reelprrecn 11217	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
25		ssidd 3954	. . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ)
26		dvmptresicc.fdv	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵))
27	26	dmeqd 5889	. . . . . . . 8 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom (𝑥 ∈ ℂ ↦ 𝐵))
28		dvmptresicc.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
29	28	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℂ 𝐵 ∈ ℂ)
30		dmmptg 6238	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℂ 𝐵 ∈ ℂ → dom (𝑥 ∈ ℂ ↦ 𝐵) = ℂ)
31	29, 30	syl 18	. . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ ℂ ↦ 𝐵) = ℂ)
32	27, 31	eqtr2d 2796	. . . . . . 7 ⊢ (𝜑 → ℂ = dom (ℂ D 𝐹))
33	7, 32	sseqtrd 3967	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹))
34		dvres3 26141	. . . . . 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 852	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
36		iccntr 25049	. . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
37	3, 4, 36	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷)) = (𝐶(,)𝐷))
38	35, 37	reseq12d 5973	. . . 4 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)))
39		ioossre 13461	. . . . 5 ⊢ (𝐶(,)𝐷) ⊆ ℝ
40		resabs1 5999	. . . . 5 ⊢ ((𝐶(,)𝐷) ⊆ ℝ → (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)) = ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)))
41	39, 40	mp1i 14	. . . 4 ⊢ (𝜑 → (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)) = ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)))
42	26	reseq1d 5971	. . . . 5 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)))
43		ioosscn 13462	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℂ
44		resmpt 6033	. . . . . 6 ⊢ ((𝐶(,)𝐷) ⊆ ℂ → ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
45	43, 44	mp1i 14	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
46	42, 45	eqtrd 2795	. . . 4 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
47	38, 41, 46	3eqtrd 2799	. . 3 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
48	14, 22, 47	3eqtrd 2799	. 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
49	11, 48	eqtr3d 2797	1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ⊆ wss 3899 {cpr 4586 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 (,)cioo 13399 [,]cicc 13402 TopOpenctopn 17507 topGenctg 17523 ℂ_fldccnfld 21586 intcnt 23243 D cdv 26091

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-map 8829 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fi 9382 df-sup 9413 df-inf 9414 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ico 13405 df-icc 13406 df-fz 13563 df-seq 14067 df-exp 14127 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-struct 17240 df-slot 17275 df-ndx 17287 df-base 17303 df-plusg 17356 df-mulr 17357 df-starv 17358 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-rest 17508 df-topn 17509 df-topgen 17529 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cnp 23454 df-haus 23541 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-limc 26094 df-dv 26095

This theorem is used by: resdvopclptsd 42895 itgsincmulx 46803

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