Theorem dvres 22078

Description: Restriction of a derivative. Note that our definition of derivative df-dv 22034 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 , +oo ) and ( -oo , 0 ]. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)

Hypotheses

Ref	Expression
dvres.k	|- K = ( TopOpen ` ℂfld )
dvres.t	|- T = ( K ↾t S )

Assertion

Ref	Expression
dvres	|- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ 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 22037	. 2 |- Rel ( S _D ( F |` B ) )
2		relres 5301	. 2 |- Rel ( ( S _D F ) |` ( ( int ` T ) ` B ) )
3		simpll 753	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> S C_ CC )
4		simplr 754	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> F : A --> CC )
5		inss1 3718	. . . . . . . 8 |- ( A i^i B ) C_ A
6		fssres 5751	. . . . . . . 8 |- ( ( F : A --> CC /\ ( A i^i B ) C_ A ) -> ( F |` ( A i^i B ) ) : ( A i^i B ) --> CC )
7	4, 5, 6	sylancl 662	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` ( A i^i B ) ) : ( A i^i B ) --> CC )
8		resres 5286	. . . . . . . . 9 |- ( ( F |` A ) |` B ) = ( F |` ( A i^i B ) )
9		ffn 5731	. . . . . . . . . . 11 |- ( F : A --> CC -> F Fn A )
10		fnresdm 5690	. . . . . . . . . . 11 |- ( F Fn A -> ( F |` A ) = F )
11	4, 9, 10	3syl 20	. . . . . . . . . 10 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` A ) = F )
12	11	reseq1d 5272	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( ( F |` A ) |` B ) = ( F |` B ) )
13	8, 12	syl5eqr 2522	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` ( A i^i B ) ) = ( F |` B ) )
14	13	feq1d 5717	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( ( F |` ( A i^i B ) ) : ( A i^i B ) --> CC <-> ( F |` B ) : ( A i^i B ) --> CC ) )
15	7, 14	mpbid 210	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` B ) : ( A i^i B ) --> CC )
16		simprl 755	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> A C_ S )
17	5, 16	syl5ss 3515	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( A i^i B ) C_ S )
18	3, 15, 17	dvcl 22066	. . . . 5 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ x ( S _D ( F |` B ) ) y ) -> y e. CC )
19	18	ex 434	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( x ( S _D ( F |` B ) ) y -> y e. CC ) )
20	3, 4, 16	dvcl 22066	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ x ( S _D F ) y ) -> y e. CC )
21	20	ex 434	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( x ( S _D F ) y -> y e. CC ) )
22	21	adantrd 468	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( ( x ( S _D F ) y /\ x e. ( ( int ` T ) ` B ) ) -> y e. CC ) )
23		dvres.k	. . . . . 6 |- K = ( TopOpen ` ℂfld )
24		dvres.t	. . . . . 6 |- T = ( K ↾t S )
25		eqid 2467	. . . . . 6 |- ( z e. ( A \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( A \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
26	3	adantr 465	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> S C_ CC )
27	4	adantr 465	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> F : A --> CC )
28	16	adantr 465	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> A C_ S )
29		simplrr 760	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> B C_ S )
30		simpr 461	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> y e. CC )
31	23, 24, 25, 26, 27, 28, 29, 30	dvreslem 22076	. . . . 5 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> ( x ( S _D ( F |` B ) ) y <-> ( x ( S _D F ) y /\ x e. ( ( int ` T ) ` B ) ) ) )
32	31	ex 434	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( y e. CC -> ( x ( S _D ( F |` B ) ) y <-> ( x ( S _D F ) y /\ x e. ( ( int ` T ) ` B ) ) ) ) )
33	19, 22, 32	pm5.21ndd 354	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( x ( S _D ( F |` B ) ) y <-> ( x ( S _D F ) y /\ x e. ( ( int ` T ) ` B ) ) ) )
34		vex 3116	. . . 4 |- y e. _V
35	34	brres 5280	. . 3 |- ( x ( ( S _D F ) |` ( ( int ` T ) ` B ) ) y <-> ( x ( S _D F ) y /\ x e. ( ( int ` T ) ` B ) ) )
36	33, 35	syl6bbr 263	. 2 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( x ( S _D ( F |` B ) ) y <-> x ( ( S _D F ) |` ( ( int ` T ) ` B ) ) y ) )
37	1, 2, 36	eqbrrdiv 5101	1 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( S _D ( F |` B ) ) = ( ( S _D F ) |` ( ( int ` T ) ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   \ cdif 3473   i^i cin 3475   C_ wss 3476   {csn 4027   class class class wbr 4447   |-> cmpt 4505   |` cres 5001   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   - cmin 9805   / cdiv 10206   ↾_t crest 14676   TopOpenctopn 14677  ℂ_fldccnfld 18219   intcnt 19312   _D cdv 22030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-fz 11673  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-rest 14678  df-topn 14679  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-cnp 19523  df-xms 20586  df-ms 20587  df-limc 22033  df-dv 22034

This theorem is referenced by:  dvcmulf  22111  dvmptres2  22128  dvmptntr  22137  dvlip  22157  dvlipcn  22158  dvlip2  22159  c1liplem1  22160  dvgt0lem1  22166  dvne0  22175  lhop1lem  22177  lhop  22180  dvcnvrelem1  22181  dvcvx  22184  ftc2ditglem  22209  pserdv  22586  efcvx  22606  dvlog  22788  dvlog2  22790  dvresntr  31274  dvmptresicc  31277  dvresioo  31279  dvbdfbdioolem1  31286  itgcoscmulx  31315  itgiccshift  31326  itgperiod  31327  dirkercncflem2  31432  fourierdlem57  31492  fourierdlem58  31493  fourierdlem72  31507  fourierdlem73  31508  fourierdlem74  31509  fourierdlem75  31510  fourierdlem80  31515  fourierdlem94  31529  fourierdlem103  31538  fourierdlem104  31539  fourierdlem113  31548