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Theorem dvres 26035

Description: Restriction of a derivative. Note that our definition of derivative df-dv 25991 would still make sense if we demanded that 𝑥 be an element of the domain instead of an interior point of the domain, but then it is possible for a non-differentiable function to have two different derivatives at a single point 𝑥 when restricted to different subsets containing 𝑥; a classic example is the absolute value function restricted to [0, +∞) and (-∞, 0]. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

**Hypotheses**
Ref	Expression
dvres.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
dvres.t	⊢ 𝑇 = (𝐾 ↾_t 𝑆)

**Assertion**
Ref	Expression
dvres	⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑆 D (𝐹 ↾ 𝐵)) = ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵)))

Proof of Theorem dvres Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldv 25994	. 2 ⊢ Rel (𝑆 D (𝐹 ↾ 𝐵))
2		relres 6002	. 2 ⊢ Rel ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))
3		simpll 778	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝑆 ⊆ ℂ)
4		simplr 780	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐹:𝐴⟶ℂ)
5		inss1 4197	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
6		fssres 6742	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
7	4, 5, 6	sylancl 597	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
8		resres 5989	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ (𝐴 ∩ 𝐵))
9		ffn 6703	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
10		fnresdm 6652	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
11	4, 9, 10	3syl 19	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ 𝐴) = 𝐹)
12	11	reseq1d 5975	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ 𝐵))
13	8, 12	eqtr3id 2818	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
14	13	feq1d 6685	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ ↔ (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ))
15	7, 14	mpbid 235	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ)
16		simprl 782	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
17	5, 16	sstrid 3956	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐴 ∩ 𝐵) ⊆ 𝑆)
18	3, 15, 17	dvcl 26023	. . . . 5 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦) → 𝑦 ∈ ℂ)
19	18	ex 417	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 → 𝑦 ∈ ℂ))
20	3, 4, 16	dvcl 26023	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
21	20	ex 417	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D 𝐹)𝑦 → 𝑦 ∈ ℂ))
22	21	adantld 495	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ))
23		dvres.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
24		dvres.t	. . . . . 6 ⊢ 𝑇 = (𝐾 ↾_t 𝑆)
25		eqid 2769	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))
26	3	adantr 485	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝑆 ⊆ ℂ)
27	4	adantr 485	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐹:𝐴⟶ℂ)
28	16	adantr 485	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐴 ⊆ 𝑆)
29		simplrr 789	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐵 ⊆ 𝑆)
30		simpr 489	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
31	23, 24, 25, 26, 27, 28, 29, 30	dvreslem 26033	. . . . 5 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦)))
32	31	ex 417	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑦 ∈ ℂ → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦))))
33	19, 22, 32	pm5.21ndd 382	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦)))
34		vex 3467	. . . 4 ⊢ 𝑦 ∈ V
35	34	brresi 5985	. . 3 ⊢ (𝑥((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦))
36	33, 35	bitr4di 292	. 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ 𝑥((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))𝑦))
37	1, 2, 36	eqbrrdiv 5778	1 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑆 D (𝐹 ↾ 𝐵)) = ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∖ cdif 3910 ∩ cin 3912 ⊆ wss 3913 {csn 4591 class class class wbr 5110 ↦ cmpt 5193 ↾ cres 5661 Fn wfn 6529 ⟶wf 6530 ‘cfv 6534 (class class class)co 7408 ℂcc 11094 − cmin 11437 / cdiv 11867 ↾_t crest 17469 TopOpenctopn 17470 ℂ_fldccnfld 21487 intcnt 23139 D cdv 25987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9367 df-sup 9398 df-inf 9399 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-fz 13532 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17203 df-slot 17238 df-ndx 17250 df-base 17266 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-rest 17471 df-topn 17472 df-topgen 17492 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-ntr 23142 df-cls 23143 df-cnp 23350 df-xms 24442 df-ms 24443 df-limc 25990 df-dv 25991

This theorem is referenced by: dvmptresicc 26040 dvcmulf 26069 dvmptres2 26086 dvmptntr 26095 dvlip 26117 dvlipcn 26118 dvlip2 26119 c1liplem1 26120 dvgt0lem1 26126 dvne0 26135 lhop1lem 26137 lhop 26140 dvcnvrelem1 26141 dvcvx 26144 ftc2ditglem 26169 pserdv 26554 efcvx 26574 dvlog 26778 dvlog2 26780 ftc2re 34926 dvun 43005 dvresntr 46519 dvresioo 46522 dvbdfbdioolem1 46529 itgcoscmulx 46570 itgiccshift 46581 itgperiod 46582 dirkercncflem2 46705 fourierdlem57 46764 fourierdlem58 46765 fourierdlem72 46779 fourierdlem73 46780 fourierdlem74 46781 fourierdlem75 46782 fourierdlem80 46787 fourierdlem94 46801 fourierdlem103 46810 fourierdlem104 46811 fourierdlem113 46820

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