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Theorem dvres 19788
Description: Restriction of a derivative. Note that our definition of derivative df-dv 19744 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 ` fld )
dvres.t  |-  T  =  ( Kt  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.
StepHypRef Expression
1 reldv 19747 . 2  |-  Rel  ( S  _D  ( F  |`  B ) )
2 relres 5166 . 2  |-  Rel  (
( S  _D  F
)  |`  ( ( int `  T ) `  B
) )
3 simpll 731 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  S  C_  CC )
4 simplr 732 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  F : A --> CC )
5 inss1 3553 . . . . . . . 8  |-  ( A  i^i  B )  C_  A
6 fssres 5602 . . . . . . . 8  |-  ( ( F : A --> CC  /\  ( A  i^i  B ) 
C_  A )  -> 
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) --> CC )
74, 5, 6sylancl 644 . . . . . . 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 5151 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  B )  =  ( F  |`  ( A  i^i  B ) )
9 ffn 5583 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  F  Fn  A )
10 fnresdm 5546 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
114, 9, 103syl 19 . . . . . . . . . 10  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  A )  =  F )
1211reseq1d 5137 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( F  |`  A )  |`  B )  =  ( F  |`  B )
)
138, 12syl5eqr 2481 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  ( A  i^i  B ) )  =  ( F  |`  B )
)
1413feq1d 5572 . . . . . . 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 ) )
157, 14mpbid 202 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  B ) : ( A  i^i  B
) --> CC )
16 simprl 733 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  A  C_  S )
175, 16syl5ss 3351 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( A  i^i  B )  C_  S )
183, 15, 17dvcl 19776 . . . . 5  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  x ( S  _D  ( F  |`  B ) ) y )  -> 
y  e.  CC )
1918ex 424 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
x ( S  _D  ( F  |`  B ) ) y  ->  y  e.  CC ) )
203, 4, 16dvcl 19776 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  x ( S  _D  F ) y )  ->  y  e.  CC )
2120ex 424 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
x ( S  _D  F ) y  -> 
y  e.  CC ) )
2221adantrd 455 . . . 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 ` fld )
24 dvres.t . . . . . 6  |-  T  =  ( Kt  S )
25 eqid 2435 . . . . . 6  |-  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
263adantr 452 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  S  C_  CC )
274adantr 452 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  F : A --> CC )
2816adantr 452 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  A  C_  S )
29 simplrr 738 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  B  C_  S )
30 simpr 448 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  y  e.  CC )
3123, 24, 25, 26, 27, 28, 29, 30dvreslem 19786 . . . . 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 )
) ) )
3231ex 424 . . . 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 )
) ) ) )
3319, 22, 32pm5.21ndd 344 . . 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 2951 . . . 4  |-  y  e. 
_V
3534brres 5144 . . 3  |-  ( x ( ( S  _D  F )  |`  (
( int `  T
) `  B )
) y  <->  ( x
( S  _D  F
) y  /\  x  e.  ( ( int `  T
) `  B )
) )
3633, 35syl6bbr 255 . 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 ) )
371, 2, 36eqbrrdiv 4966 1  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( S  _D  ( F  |`  B ) )  =  ( ( S  _D  F )  |`  (
( int `  T
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   class class class wbr 4204    e. cmpt 4258    |` cres 4872    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8978    - cmin 9281    / cdiv 9667   ↾t crest 13638   TopOpenctopn 13639  ℂfldccnfld 16693   intcnt 17071    _D cdv 19740
This theorem is referenced by:  dvcmulf  19821  dvmptres2  19838  dvmptntr  19847  dvlip  19867  dvlipcn  19868  dvlip2  19869  c1liplem1  19870  dvgt0lem1  19876  dvne0  19885  lhop1lem  19887  lhop  19890  dvcnvrelem1  19891  dvcvx  19894  ftc2ditglem  19919  pserdv  20335  efcvx  20355  dvlog  20532  dvlog2  20534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-fz 11034  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-plusg 13532  df-mulr 13533  df-starv 13534  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-rest 13640  df-topn 13641  df-topgen 13657  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cld 17073  df-ntr 17074  df-cls 17075  df-cnp 17282  df-xms 18340  df-ms 18341  df-limc 19743  df-dv 19744
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