dvmptresicc - Metamath Proof Explorer

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Theorem dvmptresicc 26236

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 5966	. . . 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 13565	. . . . . 6 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
6		ax-resscn 11257	. . . . . . 7 ⊢ ℝ ⊆ ℂ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
8	5, 7	sstrd 3941	. . . . 5 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℂ)
9	8	resmptd 6032	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ 𝐴) ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))
10	2, 9	eqtrid 2808	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴))
11	10	oveq2d 7436	. 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)))
12	5	resabs1d 5999	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)) = (𝐹 ↾ (𝐶[,]𝐷)))
13	12	eqcomd 2767	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐶[,]𝐷)) = ((𝐹 ↾ ℝ) ↾ (𝐶[,]𝐷)))
14	13	oveq2d 7436	. . 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 7114	. . . . 5 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
17	16, 7	fssresd 6749	. . . 4 ⊢ (𝜑 → (𝐹 ↾ ℝ):ℝ⟶ℂ)
18		ssidd 3954	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ)
19		eqid 2761	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20		tgioo4 25124	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21	19, 20	dvres 26231	. . . 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 11292	. . . . . . 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 5887	. . . . . . . 8 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom (𝑥 ∈ ℂ ↦ 𝐵))
28		dvmptresicc.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
29	28	ralrimiva 3155	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ ℂ 𝐵 ∈ ℂ)
30		dmmptg 6243	. . . . . . . . 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 2797	. . . . . . 7 ⊢ (𝜑 → ℂ = dom (ℂ D 𝐹))
33	7, 32	sseqtrd 3967	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹))
34		dvres3 26233	. . . . . 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 25141	. . . . . 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 5971	. . . 4 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (((ℂ D 𝐹) ↾ ℝ) ↾ (𝐶(,)𝐷)))
39		ioossre 13538	. . . . 5 ⊢ (𝐶(,)𝐷) ⊆ ℝ
40		resabs1 5997	. . . . 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 5969	. . . . 5 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = ((𝑥 ∈ ℂ ↦ 𝐵) ↾ (𝐶(,)𝐷)))
43		ioosscn 13539	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℂ
44		resmpt 6029	. . . . . 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 2796	. . . 4 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ (𝐶(,)𝐷)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
47	38, 41, 46	3eqtrd 2800	. . 3 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ ℝ)) ↾ ((int‘(topGen‘ran (,)))‘(𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
48	14, 22, 47	3eqtrd 2800	. 2 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐶[,]𝐷))) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
49	11, 48	eqtr3d 2798	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 3077 ⊆ wss 3899 {cpr 4586 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 (,)cioo 13476 [,]cicc 13479 TopOpenctopn 17592 topGenctg 17608 ℂ_fldccnfld 21678 intcnt 23335 D cdv 26183

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ico 13482 df-icc 13483 df-fz 13640 df-seq 14145 df-exp 14205 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-struct 17325 df-slot 17360 df-ndx 17372 df-base 17388 df-plusg 17441 df-mulr 17442 df-starv 17443 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-rest 17593 df-topn 17594 df-topgen 17614 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cnp 23546 df-haus 23633 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-limc 26186 df-dv 26187

This theorem is used by: resdvopclptsd 43078 itgsincmulx 46983

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