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Theorem dvres 25880

Description: Restriction of a derivative. Note that our definition of derivative df-dv 25836 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 25839	. 2 ⊢ Rel (𝑆 D (𝐹 ↾ 𝐵))
2		relres 5972	. 2 ⊢ Rel ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))
3		simpll 767	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝑆 ⊆ ℂ)
4		simplr 769	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐹:𝐴⟶ℂ)
5		inss1 4191	. . . . . . . 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 587	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)⟶ℂ)
8		resres 5959	. . . . . . . . 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 5945	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝐹 ↾ 𝐴) ↾ 𝐵) = (𝐹 ↾ 𝐵))
13	8, 12	eqtr3id 2786	. . . . . . . 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 771	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
17	5, 16	sstrid 3947	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝐴 ∩ 𝐵) ⊆ 𝑆)
18	3, 15, 17	dvcl 25868	. . . . 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 25868	. . . . . 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 2737	. . . . . 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 778	. . . . . 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 25878	. . . . 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 3446	. . . 4 ⊢ 𝑦 ∈ V
35	34	brresi 5955	. . 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 5751	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 1542 ∈ wcel 2114 ∖ cdif 3900 ∩ cin 3902 ⊆ wss 3903 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 ↾ cres 5634 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 − cmin 11376 / cdiv 11806 ↾_t crest 17352 TopOpenctopn 17353 ℂ_fldccnfld 21321 intcnt 22973 D cdv 25832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-fz 13436 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-rest 17354 df-topn 17355 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-cnp 23184 df-xms 24276 df-ms 24277 df-limc 25835 df-dv 25836

This theorem is referenced by: dvmptresicc 25885 dvcmulf 25916 dvmptres2 25934 dvmptntr 25943 dvlip 25966 dvlipcn 25967 dvlip2 25968 c1liplem1 25969 dvgt0lem1 25975 dvne0 25984 lhop1lem 25986 lhop 25989 dvcnvrelem1 25990 dvcvx 25993 ftc2ditglem 26020 pserdv 26407 efcvx 26427 dvlog 26628 dvlog2 26630 ftc2re 34776 dvun 42729 dvresntr 46276 dvresioo 46279 dvbdfbdioolem1 46286 itgcoscmulx 46327 itgiccshift 46338 itgperiod 46339 dirkercncflem2 46462 fourierdlem57 46521 fourierdlem58 46522 fourierdlem72 46536 fourierdlem73 46537 fourierdlem74 46538 fourierdlem75 46539 fourierdlem80 46544 fourierdlem94 46558 fourierdlem103 46567 fourierdlem104 46568 fourierdlem113 46577

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