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Theorem dvres 25788

Description: Restriction of a derivative. Note that our definition of derivative df-dv 25744 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 25747	. 2 ⊢ Rel (𝑆 D (𝐹 ↾ 𝐵))
2		relres 5965	. 2 ⊢ Rel ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))
3		simpll 766	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝑆 ⊆ ℂ)
4		simplr 768	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐹:𝐴⟶ℂ)
5		inss1 4196	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
6		fssres 6708	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
7	4, 5, 6	sylancl 586	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
8		resres 5952	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ (𝐴 ∩ 𝐵))
9		ffn 6670	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
10		fnresdm 6619	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
11	4, 9, 10	3syl 18	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ 𝐴) = 𝐹)
12	11	reseq1d 5938	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ 𝐵))
13	8, 12	eqtr3id 2778	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
14	13	feq1d 6652	. . . . . . 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 232	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ)
16		simprl 770	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
17	5, 16	sstrid 3955	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐴 ∩ 𝐵) ⊆ 𝑆)
18	3, 15, 17	dvcl 25776	. . . . 5 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦) → 𝑦 ∈ ℂ)
19	18	ex 412	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 → 𝑦 ∈ ℂ))
20	3, 4, 16	dvcl 25776	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ)
21	20	ex 412	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑥(𝑆 D 𝐹)𝑦 → 𝑦 ∈ ℂ))
22	21	adantld 490	. . . 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 2729	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)))
26	3	adantr 480	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝑆 ⊆ ℂ)
27	4	adantr 480	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐹:𝐴⟶ℂ)
28	16	adantr 480	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐴 ⊆ 𝑆)
29		simplrr 777	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝐵 ⊆ 𝑆)
30		simpr 484	. . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
31	23, 24, 25, 26, 27, 28, 29, 30	dvreslem 25786	. . . . 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 412	. . . 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 379	. . 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 3448	. . . 4 ⊢ 𝑦 ∈ V
35	34	brresi 5948	. . 3 ⊢ (𝑥((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦))
36	33, 35	bitr4di 289	. 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 5748	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3908 ∩ cin 3910 ⊆ wss 3911 {csn 4585 class class class wbr 5102 ↦ cmpt 5183 ↾ cres 5633 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 − cmin 11381 / cdiv 11811 ↾_t crest 17359 TopOpenctopn 17360 ℂ_fldccnfld 21240 intcnt 22880 D cdv 25740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-fz 13445 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17361 df-topn 17362 df-topgen 17382 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-cnp 23091 df-xms 24184 df-ms 24185 df-limc 25743 df-dv 25744

This theorem is referenced by: dvmptresicc 25793 dvcmulf 25824 dvmptres2 25842 dvmptntr 25851 dvlip 25874 dvlipcn 25875 dvlip2 25876 c1liplem1 25877 dvgt0lem1 25883 dvne0 25892 lhop1lem 25894 lhop 25897 dvcnvrelem1 25898 dvcvx 25901 ftc2ditglem 25928 pserdv 26315 efcvx 26335 dvlog 26536 dvlog2 26538 ftc2re 34562 dvun 42320 dvresntr 45889 dvresioo 45892 dvbdfbdioolem1 45899 itgcoscmulx 45940 itgiccshift 45951 itgperiod 45952 dirkercncflem2 46075 fourierdlem57 46134 fourierdlem58 46135 fourierdlem72 46149 fourierdlem73 46150 fourierdlem74 46151 fourierdlem75 46152 fourierdlem80 46157 fourierdlem94 46171 fourierdlem103 46180 fourierdlem104 46181 fourierdlem113 46190

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