Theorem dvres 22945

Description: Restriction of a derivative. Note that our definition of derivative df-dv 22901 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 22904	. 2 |- Rel ( S _D ( F |` B ) )
2		relres 5138	. 2 |- Rel ( ( S _D F ) |` ( ( int ` T ) ` B ) )
3		simpll 768	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> S C_ CC )
4		simplr 770	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> F : A --> CC )
5		inss1 3643	. . . . . . . 8 |- ( A i^i B ) C_ A
6		fssres 5761	. . . . . . . 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 675	. . . . . . 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 5123	. . . . . . . . 9 |- ( ( F |` A ) |` B ) = ( F |` ( A i^i B ) )
9		ffn 5739	. . . . . . . . . . 11 |- ( F : A --> CC -> F Fn A )
10		fnresdm 5695	. . . . . . . . . . 11 |- ( F Fn A -> ( F |` A ) = F )
11	4, 9, 10	3syl 18	. . . . . . . . . 10 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` A ) = F )
12	11	reseq1d 5110	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( ( F |` A ) |` B ) = ( F |` B ) )
13	8, 12	syl5eqr 2519	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` ( A i^i B ) ) = ( F |` B ) )
14	13	feq1d 5724	. . . . . . 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 215	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` B ) : ( A i^i B ) --> CC )
16		simprl 772	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> A C_ S )
17	5, 16	syl5ss 3429	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( A i^i B ) C_ S )
18	3, 15, 17	dvcl 22933	. . . . 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 441	. . . 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 22933	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ x ( S _D F ) y ) -> y e. CC )
21	20	ex 441	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( x ( S _D F ) y -> y e. CC ) )
22	21	adantrd 475	. . . 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 2471	. . . . . 6 |- ( z e. ( A \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( A \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
26	3	adantr 472	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> S C_ CC )
27	4	adantr 472	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> F : A --> CC )
28	16	adantr 472	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> A C_ S )
29		simplrr 779	. . . . . 6 |- ( ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) /\ y e. CC ) -> B C_ S )
30		simpr 468	. . . . . 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 22943	. . . . 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 441	. . . 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 361	. . 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 3034	. . . 4 |- y e. _V
35	34	brres 5117	. . 3 |- ( x ( ( S _D F ) |` ( ( int ` T ) ` B ) ) y <-> ( x ( S _D F ) y /\ x e. ( ( int ` T ) ` B ) ) )
36	33, 35	syl6bbr 271	. 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 4938	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   \ cdif 3387   i^i cin 3389   C_ wss 3390   {csn 3959   class class class wbr 4395   |-> cmpt 4454   |` cres 4841   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   - cmin 9880   / cdiv 10291   ↾_t crest 15397   TopOpenctopn 15398  ℂ_fldccnfld 19047   intcnt 20109   _D cdv 22897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-cnp 20321  df-xms 21413  df-ms 21414  df-limc 22900  df-dv 22901

This theorem is referenced by:  dvcmulf  22978  dvmptres2  22995  dvmptntr  23004  dvlip  23024  dvlipcn  23025  dvlip2  23026  c1liplem1  23027  dvgt0lem1  23033  dvne0  23042  lhop1lem  23044  lhop  23047  dvcnvrelem1  23048  dvcvx  23051  ftc2ditglem  23076  pserdv  23463  efcvx  23483  dvlog  23675  dvlog2  23677  dvresntr  37885  dvmptresicc  37888  dvresioo  37890  dvbdfbdioolem1  37897  itgcoscmulx  37943  itgiccshift  37954  itgperiod  37955  dirkercncflem2  38078  fourierdlem57  38139  fourierdlem58  38140  fourierdlem72  38154  fourierdlem73  38155  fourierdlem74  38156  fourierdlem75  38157  fourierdlem80  38162  fourierdlem94  38176  fourierdlem103  38185  fourierdlem104  38186  fourierdlem113  38195