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Theorem dvres 25868

Description: Restriction of a derivative. Note that our definition of derivative df-dv 25824 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 25827	. 2 ⊢ Rel (𝑆 D (𝐹 ↾ 𝐵))
2		relres 5964	. 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 4189	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
6		fssres 6700	. . . . . . . 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 5951	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ (𝐴 ∩ 𝐵))
9		ffn 6662	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
10		fnresdm 6611	. . . . . . . . . . 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 5937	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ 𝐵))
13	8, 12	eqtr3id 2785	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐹 ↾ 𝐵))
14	13	feq1d 6644	. . . . . . 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 3945	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐴 ∩ 𝐵) ⊆ 𝑆)
18	3, 15, 17	dvcl 25856	. . . . 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 25856	. . . . . 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 2736	. . . . . 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 25866	. . . . 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 3444	. . . 4 ⊢ 𝑦 ∈ V
35	34	brresi 5947	. . 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 5743	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 1541 ∈ wcel 2113 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 {csn 4580 class class class wbr 5098 ↦ cmpt 5179 ↾ cres 5626 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 − cmin 11364 / cdiv 11794 ↾_t crest 17340 TopOpenctopn 17341 ℂ_fldccnfld 21309 intcnt 22961 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-fz 13424 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-slot 17109 df-ndx 17121 df-base 17137 df-plusg 17190 df-mulr 17191 df-starv 17192 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-rest 17342 df-topn 17343 df-topgen 17363 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-cnp 23172 df-xms 24264 df-ms 24265 df-limc 25823 df-dv 25824

This theorem is referenced by: dvmptresicc 25873 dvcmulf 25904 dvmptres2 25922 dvmptntr 25931 dvlip 25954 dvlipcn 25955 dvlip2 25956 c1liplem1 25957 dvgt0lem1 25963 dvne0 25972 lhop1lem 25974 lhop 25977 dvcnvrelem1 25978 dvcvx 25981 ftc2ditglem 26008 pserdv 26395 efcvx 26415 dvlog 26616 dvlog2 26618 ftc2re 34755 dvun 42614 dvresntr 46162 dvresioo 46165 dvbdfbdioolem1 46172 itgcoscmulx 46213 itgiccshift 46224 itgperiod 46225 dirkercncflem2 46348 fourierdlem57 46407 fourierdlem58 46408 fourierdlem72 46422 fourierdlem73 46423 fourierdlem74 46424 fourierdlem75 46425 fourierdlem80 46430 fourierdlem94 46444 fourierdlem103 46453 fourierdlem104 46454 fourierdlem113 46463

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