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Theorem dvfre 22220
Description: The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)
Assertion
Ref Expression
dvfre  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )

Proof of Theorem dvfre
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvf 22177 . . 3  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
2 ffn 5717 . . 3  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
31, 2mp1i 12 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
41ffvelrni 6011 . . . . 5  |-  ( x  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  x
)  e.  CC )
54adantl 466 . . . 4  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  CC )
6 simpr 461 . . . . . 6  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
7 fvco3 5931 . . . . . 6  |-  ( ( ( RR  _D  F
) : dom  ( RR  _D  F ) --> CC 
/\  x  e.  dom  ( RR  _D  F
) )  ->  (
( *  o.  ( RR  _D  F ) ) `
 x )  =  ( * `  (
( RR  _D  F
) `  x )
) )
81, 6, 7sylancr 663 . . . . 5  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( *  o.  ( RR  _D  F ) ) `
 x )  =  ( * `  (
( RR  _D  F
) `  x )
) )
9 ax-resscn 9547 . . . . . . . . . 10  |-  RR  C_  CC
10 fss 5725 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
119, 10mpan2 671 . . . . . . . . 9  |-  ( F : A --> RR  ->  F : A --> CC )
12 dvcj 22219 . . . . . . . . 9  |-  ( ( F : A --> CC  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
1311, 12sylan 471 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
14 ffvelrn 6010 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  y  e.  A )  ->  ( F `  y
)  e.  RR )
1514adantlr 714 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  RR )
1615cjred 13033 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( * `  ( F `  y
) )  =  ( F `  y ) )
1716mpteq2dva 4519 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( y  e.  A  |->  ( * `  ( F `  y )
) )  =  ( y  e.  A  |->  ( F `  y ) ) )
1815recnd 9620 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  CC )
19 simpl 457 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F : A --> RR )
2019feqmptd 5907 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
21 cjf 12911 . . . . . . . . . . . . 13  |-  * : CC --> CC
2221a1i 11 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  * : CC --> CC )
2322feqmptd 5907 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  *  =  ( z  e.  CC  |->  ( * `  z ) ) )
24 fveq2 5852 . . . . . . . . . . 11  |-  ( z  =  ( F `  y )  ->  (
* `  z )  =  ( * `  ( F `  y ) ) )
2518, 20, 23, 24fmptco 6045 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  F
)  =  ( y  e.  A  |->  ( * `
 ( F `  y ) ) ) )
2617, 25, 203eqtr4d 2492 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  F
)  =  F )
2726oveq2d 6293 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( RR 
_D  F ) )
2813, 27eqtr3d 2484 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( RR  _D  F ) )
2928fveq1d 5854 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( *  o.  ( RR  _D  F
) ) `  x
)  =  ( ( RR  _D  F ) `
 x ) )
3029adantr 465 . . . . 5  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( *  o.  ( RR  _D  F ) ) `
 x )  =  ( ( RR  _D  F ) `  x
) )
318, 30eqtr3d 2484 . . . 4  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
* `  ( ( RR  _D  F ) `  x ) )  =  ( ( RR  _D  F ) `  x
) )
325, 31cjrebd 13009 . . 3  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  RR )
3332ralrimiva 2855 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  A. x  e.  dom  ( RR  _D  F
) ( ( RR 
_D  F ) `  x )  e.  RR )
34 ffnfv 6038 . 2  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> RR  <->  ( ( RR  _D  F
)  Fn  dom  ( RR  _D  F )  /\  A. x  e.  dom  ( RR  _D  F ) ( ( RR  _D  F
) `  x )  e.  RR ) )
353, 33, 34sylanbrc 664 1  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458    |-> cmpt 4491   dom cdm 4985    o. ccom 4989    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   *ccj 12903    _D cdv 22133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fi 7869  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-icc 11540  df-fz 11677  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-plusg 14582  df-mulr 14583  df-starv 14584  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-rest 14692  df-topn 14693  df-topgen 14713  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-cncf 21248  df-limc 22136  df-dv 22137
This theorem is referenced by:  dvnfre  22221  dvferm1lem  22251  dvferm1  22252  dvferm2lem  22253  dvferm2  22254  dvferm  22255  c1lip2  22265  dvle  22274  dvivthlem1  22275  dvivth  22277  dvne0  22278  dvfsumle  22288  dvfsumge  22289  dvmptrecl  22291  dvbdfbdioolem1  31625  dvbdfbdioolem2  31626  ioodvbdlimc1lem1  31628  ioodvbdlimc1lem2  31629  ioodvbdlimc2lem  31631  fourierdlem58  31832  fourierdlem59  31833  fourierdlem60  31834  fourierdlem61  31835  fourierdlem94  31868  fourierdlem97  31871  fourierdlem112  31886  fourierdlem113  31887
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