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Theorem dvres 26121

Description: Restriction of a derivative. Note that our definition of derivative df-dv 26077 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 26080	. 2 ⊢ Rel (𝑆 D (𝐹 ↾ 𝐵))
2		relres 6006	. 2 ⊢ Rel ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))
3		simpll 779	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝑆 ⊆ ℂ)
4		simplr 781	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐹:𝐴⟶ℂ)
5		inss1 4189	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
6		fssres 6748	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
7	4, 5, 6	sylancl 598	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
8		resres 5993	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ (𝐴 ∩ 𝐵))
9		ffn 6709	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
10		fnresdm 6658	. . . . . . . . . . 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 5979	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ 𝐵))
13	8, 12	eqtr3id 2814	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
14	13	feq1d 6691	. . . . . . 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 783	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
17	5, 16	sstrid 3949	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐴 ∩ 𝐵) ⊆ 𝑆)
18	3, 15, 17	dvcl 26109	. . . . 5 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦) → 𝑦 ∈ ℂ)
19	18	ex 418	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 → 𝑦 ∈ ℂ))
20	3, 4, 16	dvcl 26109	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
21	20	ex 418	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D 𝐹)𝑦 → 𝑦 ∈ ℂ))
22	21	adantld 496	. . . 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 2765	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))
26	3	adantr 486	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝑆 ⊆ ℂ)
27	4	adantr 486	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐹:𝐴⟶ℂ)
28	16	adantr 486	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐴 ⊆ 𝑆)
29		simplrr 790	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐵 ⊆ 𝑆)
30		simpr 490	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
31	23, 24, 25, 26, 27, 28, 29, 30	dvreslem 26119	. . . . 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 418	. . . 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 3461	. . . 4 ⊢ 𝑦 ∈ V
35	34	brresi 5989	. . 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 5782	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 {csn 4591 class class class wbr 5111 ↦ cmpt 5194 ↾ cres 5665 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 − cmin 11456 / cdiv 11886 ↾_t crest 17495 TopOpenctopn 17496 ℂ_fldccnfld 21572 intcnt 23224 D cdv 26073

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fi 9378 df-sup 9409 df-inf 9410 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-fz 13552 df-seq 14056 df-exp 14116 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-struct 17229 df-slot 17264 df-ndx 17276 df-base 17292 df-plusg 17345 df-mulr 17346 df-starv 17347 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-rest 17497 df-topn 17498 df-topgen 17518 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-cnp 23435 df-xms 24528 df-ms 24529 df-limc 26076 df-dv 26077

This theorem is used by: dvmptresicc 26126 dvcmulf 26155 dvmptres2 26172 dvmptntr 26181 dvlip 26203 dvlipcn 26204 dvlip2 26205 c1liplem1 26206 dvgt0lem1 26212 dvne0 26221 lhop1lem 26223 lhop 26226 dvcnvrelem1 26227 dvcvx 26230 ftc2ditglem 26255 pserdv 26643 efcvx 26663 dvlog 26867 dvlog2 26869 ftc2re 35050 dvun 43178 dvresntr 46690 dvresioo 46693 dvbdfbdioolem1 46700 itgcoscmulx 46741 itgiccshift 46752 itgperiod 46753 dirkercncflem2 46876 fourierdlem57 46935 fourierdlem58 46936 fourierdlem72 46950 fourierdlem73 46951 fourierdlem74 46952 fourierdlem75 46953 fourierdlem80 46958 fourierdlem94 46972 fourierdlem103 46981 fourierdlem104 46982 fourierdlem113 46991

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