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Theorem dvres 21408
Description: Restriction of a derivative. Note that our definition of derivative df-dv 21364 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 21367 . 2  |-  Rel  ( S  _D  ( F  |`  B ) )
2 relres 5159 . 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 754 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  F : A --> CC )
5 inss1 3591 . . . . . . . 8  |-  ( A  i^i  B )  C_  A
6 fssres 5599 . . . . . . . 8  |-  ( ( F : A --> CC  /\  ( A  i^i  B ) 
C_  A )  -> 
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) --> CC )
74, 5, 6sylancl 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 5144 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  B )  =  ( F  |`  ( A  i^i  B ) )
9 ffn 5580 . . . . . . . . . . 11  |-  ( F : A --> CC  ->  F  Fn  A )
10 fnresdm 5541 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
114, 9, 103syl 20 . . . . . . . . . 10  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  A )  =  F )
1211reseq1d 5130 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( F  |`  A )  |`  B )  =  ( F  |`  B )
)
138, 12syl5eqr 2489 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  ( A  i^i  B ) )  =  ( F  |`  B )
)
1413feq1d 5567 . . . . . . 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 210 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( F  |`  B ) : ( A  i^i  B
) --> CC )
16 simprl 755 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  A  C_  S )
175, 16syl5ss 3388 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( A  i^i  B )  C_  S )
183, 15, 17dvcl 21396 . . . . 5  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  x ( S  _D  ( F  |`  B ) ) y )  -> 
y  e.  CC )
1918ex 434 . . . 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 21396 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  x ( S  _D  F ) y )  ->  y  e.  CC )
2120ex 434 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
x ( S  _D  F ) y  -> 
y  e.  CC ) )
2221adantrd 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 ` fld )
24 dvres.t . . . . . 6  |-  T  =  ( Kt  S )
25 eqid 2443 . . . . . 6  |-  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
263adantr 465 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  S  C_  CC )
274adantr 465 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  F : A --> CC )
2816adantr 465 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  /\  y  e.  CC )  ->  A  C_  S )
29 simplrr 760 . . . . . 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 )
3123, 24, 25, 26, 27, 28, 29, 30dvreslem 21406 . . . . 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 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 )
) ) ) )
3319, 22, 32pm5.21ndd 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 2996 . . . 4  |-  y  e. 
_V
3534brres 5138 . . 3  |-  ( x ( ( S  _D  F )  |`  (
( int `  T
) `  B )
) y  <->  ( x
( S  _D  F
) y  /\  x  e.  ( ( int `  T
) `  B )
) )
3633, 35syl6bbr 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 ) )
371, 2, 36eqbrrdiv 4959 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3346    i^i cin 3348    C_ wss 3349   {csn 3898   class class class wbr 4313    e. cmpt 4371    |` cres 4863    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   CCcc 9301    - cmin 9616    / cdiv 10014   ↾t crest 14380   TopOpenctopn 14381  ℂfldccnfld 17840   intcnt 18643    _D cdv 21360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fi 7682  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-topgen 14403  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-cnp 18854  df-xms 19917  df-ms 19918  df-limc 21363  df-dv 21364
This theorem is referenced by:  dvcmulf  21441  dvmptres2  21458  dvmptntr  21467  dvlip  21487  dvlipcn  21488  dvlip2  21489  c1liplem1  21490  dvgt0lem1  21496  dvne0  21505  lhop1lem  21507  lhop  21510  dvcnvrelem1  21511  dvcvx  21514  ftc2ditglem  21539  pserdv  21916  efcvx  21936  dvlog  22118  dvlog2  22120
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