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Theorem dvres 22866
Description: Restriction of a derivative. Note that our definition of derivative df-dv 22822 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 22825 . 2  |-  Rel  ( S  _D  ( F  |`  B ) )
2 relres 5132 . 2  |-  Rel  (
( S  _D  F
)  |`  ( ( int `  T ) `  B
) )
3 simpll 760 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  S  C_  CC )
4 simplr 762 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  F : A --> CC )
5 inss1 3652 . . . . . . . 8  |-  ( A  i^i  B )  C_  A
6 fssres 5749 . . . . . . . 8  |-  ( ( F : A --> CC  /\  ( A  i^i  B ) 
C_  A )  -> 
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) --> CC )
74, 5, 6sylancl 668 . . . . . . 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 5117 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  B )  =  ( F  |`  ( A  i^i  B ) )
9 ffn 5728 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  F  Fn  A )
10 fnresdm 5685 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
114, 9, 103syl 18 . . . . . . . . . 10  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  A )  =  F )
1211reseq1d 5104 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( F  |`  A )  |`  B )  =  ( F  |`  B )
)
138, 12syl5eqr 2499 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  ( A  i^i  B ) )  =  ( F  |`  B )
)
1413feq1d 5714 . . . . . . 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 214 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  B ) : ( A  i^i  B
) --> CC )
16 simprl 764 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  A  C_  S )
175, 16syl5ss 3443 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( A  i^i  B )  C_  S )
183, 15, 17dvcl 22854 . . . . 5  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  x ( S  _D  ( F  |`  B ) ) y )  -> 
y  e.  CC )
1918ex 436 . . . 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 22854 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  x ( S  _D  F ) y )  ->  y  e.  CC )
2120ex 436 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
x ( S  _D  F ) y  -> 
y  e.  CC ) )
2221adantrd 470 . . . 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 2451 . . . . . 6  |-  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
263adantr 467 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  S  C_  CC )
274adantr 467 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  F : A --> CC )
2816adantr 467 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  A  C_  S )
29 simplrr 771 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  B  C_  S )
30 simpr 463 . . . . . 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 22864 . . . . 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 436 . . . 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 356 . . 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 3048 . . . 4  |-  y  e. 
_V
3534brres 5111 . . 3  |-  ( x ( ( S  _D  F )  |`  (
( int `  T
) `  B )
) y  <->  ( x
( S  _D  F
) y  /\  x  e.  ( ( int `  T
) `  B )
) )
3633, 35syl6bbr 267 . 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 4933 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    \ cdif 3401    i^i cin 3403    C_ wss 3404   {csn 3968   class class class wbr 4402    |-> cmpt 4461    |` cres 4836    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537    - cmin 9860    / cdiv 10269   ↾t crest 15319   TopOpenctopn 15320  ℂfldccnfld 18970   intcnt 20032    _D cdv 22818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-cnp 20244  df-xms 21335  df-ms 21336  df-limc 22821  df-dv 22822
This theorem is referenced by:  dvcmulf  22899  dvmptres2  22916  dvmptntr  22925  dvlip  22945  dvlipcn  22946  dvlip2  22947  c1liplem1  22948  dvgt0lem1  22954  dvne0  22963  lhop1lem  22965  lhop  22968  dvcnvrelem1  22969  dvcvx  22972  ftc2ditglem  22997  pserdv  23384  efcvx  23404  dvlog  23596  dvlog2  23598  dvresntr  37788  dvmptresicc  37791  dvresioo  37793  dvbdfbdioolem1  37800  itgcoscmulx  37846  itgiccshift  37857  itgperiod  37858  dirkercncflem2  37966  fourierdlem57  38027  fourierdlem58  38028  fourierdlem72  38042  fourierdlem73  38043  fourierdlem74  38044  fourierdlem75  38045  fourierdlem80  38050  fourierdlem94  38064  fourierdlem103  38073  fourierdlem104  38074  fourierdlem113  38083
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