Theorem dvres 22441

Description: Restriction of a derivative. Note that our definition of derivative df-dv 22397 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 22400	. 2 |- Rel ( S _D ( F |` B ) )
2		relres 5311	. 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 755	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> F : A --> CC )
5		inss1 3714	. . . . . . . 8 |- ( A i^i B ) C_ A
6		fssres 5757	. . . . . . . 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 5296	. . . . . . . . 9 |- ( ( F |` A ) |` B ) = ( F |` ( A i^i B ) )
9		ffn 5737	. . . . . . . . . . 11 |- ( F : A --> CC -> F Fn A )
10		fnresdm 5696	. . . . . . . . . . 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 5282	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( ( F |` A ) |` B ) = ( F |` B ) )
13	8, 12	syl5eqr 2512	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( F |` ( A i^i B ) ) = ( F |` B ) )
14	13	feq1d 5723	. . . . . . 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 756	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> A C_ S )
17	5, 16	syl5ss 3510	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( A i^i B ) C_ S )
18	3, 15, 17	dvcl 22429	. . . . 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 22429	. . . . . 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 2457	. . . . . 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 762	. . . . . 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 22439	. . . . 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 3112	. . . 4 |- y e. _V
35	34	brres 5290	. . 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 5110	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 1395   e. wcel 1819   \ cdif 3468   i^i cin 3470   C_ wss 3471   {csn 4032   class class class wbr 4456   |-> cmpt 4515   |` cres 5010   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   - cmin 9824   / cdiv 10227   ↾_t crest 14838   TopOpenctopn 14839  ℂ_fldccnfld 18547   intcnt 19645   _D cdv 22393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-rest 14840  df-topn 14841  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-cnp 19856  df-xms 20949  df-ms 20950  df-limc 22396  df-dv 22397

This theorem is referenced by:  dvcmulf  22474  dvmptres2  22491  dvmptntr  22500  dvlip  22520  dvlipcn  22521  dvlip2  22522  c1liplem1  22523  dvgt0lem1  22529  dvne0  22538  lhop1lem  22540  lhop  22543  dvcnvrelem1  22544  dvcvx  22547  ftc2ditglem  22572  pserdv  22950  efcvx  22970  dvlog  23158  dvlog2  23160  dvresntr  31916  dvmptresicc  31919  dvresioo  31921  dvbdfbdioolem1  31928  itgcoscmulx  31971  itgiccshift  31982  itgperiod  31983  dirkercncflem2  32089  fourierdlem57  32149  fourierdlem58  32150  fourierdlem72  32164  fourierdlem73  32165  fourierdlem74  32166  fourierdlem75  32167  fourierdlem80  32172  fourierdlem94  32186  fourierdlem103  32195  fourierdlem104  32196  fourierdlem113  32205