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Theorem dvres 19667
Description: Restriction of a derivative. Note that our definition of derivative df-dv 19623 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 19626 . 2  |-  Rel  ( S  _D  ( F  |`  B ) )
2 relres 5116 . 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 3506 . . . . . . . 8  |-  ( A  i^i  B )  C_  A
6 fssres 5552 . . . . . . . 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 5101 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  B )  =  ( F  |`  ( A  i^i  B ) )
9 ffn 5533 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  F  Fn  A )
10 fnresdm 5496 . . . . . . . . . . 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 5087 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( F  |`  A )  |`  B )  =  ( F  |`  B )
)
138, 12syl5eqr 2435 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  ( A  i^i  B ) )  =  ( F  |`  B )
)
1413feq1d 5522 . . . . . . 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 3304 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( A  i^i  B )  C_  S )
183, 15, 17dvcl 19655 . . . . 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 19655 . . . . . 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 2389 . . . . . 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 19665 . . . . 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 2904 . . . 4  |-  y  e. 
_V
3534brres 5094 . . 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 4916 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 1649    e. wcel 1717    \ cdif 3262    i^i cin 3264    C_ wss 3265   {csn 3759   class class class wbr 4155    e. cmpt 4209    |` cres 4822    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923    - cmin 9225    / cdiv 9611   ↾t crest 13577   TopOpenctopn 13578  ℂfldccnfld 16628   intcnt 17006    _D cdv 19619
This theorem is referenced by:  dvcmulf  19700  dvmptres2  19717  dvmptntr  19726  dvlip  19746  dvlipcn  19747  dvlip2  19748  c1liplem1  19749  dvgt0lem1  19755  dvne0  19764  lhop1lem  19766  lhop  19769  dvcnvrelem1  19770  dvcvx  19773  ftc2ditglem  19798  pserdv  20214  efcvx  20234  dvlog  20411  dvlog2  20413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-plusg 13471  df-mulr 13472  df-starv 13473  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-rest 13579  df-topn 13580  df-topgen 13596  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-ntr 17009  df-cls 17010  df-cnp 17216  df-xms 18261  df-ms 18262  df-limc 19622  df-dv 19623
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