MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvfre Unicode version

Theorem dvfre 19294
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 )
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem dvfre
StepHypRef Expression
1 dvf 19251 . . 3  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
2 ffn 5354 . . 3  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
31, 2mp1i 13 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
41ffvelrni 5625 . . . . 5  |-  ( x  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  x
)  e.  CC )
54adantl 454 . . . 4  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  CC )
6 simpr 449 . . . . . 6  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
7 fvco3 5557 . . . . . 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 646 . . . . 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 8789 . . . . . . . . . 10  |-  RR  C_  CC
10 fss 5362 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
119, 10mpan2 654 . . . . . . . . 9  |-  ( F : A --> RR  ->  F : A --> CC )
12 dvcj 19293 . . . . . . . . 9  |-  ( ( F : A --> CC  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
1311, 12sylan 459 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
14 ffvelrn 5624 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  y  e.  A )  ->  ( F `  y
)  e.  RR )
1514adantlr 697 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  RR )
1615cjred 11705 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( * `  ( F `  y
) )  =  ( F `  y ) )
1716mpteq2dva 4107 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( y  e.  A  |->  ( * `  ( F `  y )
) )  =  ( y  e.  A  |->  ( F `  y ) ) )
1815recnd 8856 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  CC )
19 simpl 445 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F : A --> RR )
2019feqmptd 5536 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
21 cjf 11583 . . . . . . . . . . . . 13  |-  * : CC --> CC
2221a1i 12 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  * : CC --> CC )
2322feqmptd 5536 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  *  =  ( z  e.  CC  |->  ( * `  z ) ) )
24 fveq2 5485 . . . . . . . . . . 11  |-  ( z  =  ( F `  y )  ->  (
* `  z )  =  ( * `  ( F `  y ) ) )
2518, 20, 23, 24fmptco 5652 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  F
)  =  ( y  e.  A  |->  ( * `
 ( F `  y ) ) ) )
2617, 25, 203eqtr4d 2326 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  F
)  =  F )
2726oveq2d 5835 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( RR 
_D  F ) )
2813, 27eqtr3d 2318 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( RR  _D  F ) )
2928fveq1d 5487 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( *  o.  ( RR  _D  F
) ) `  x
)  =  ( ( RR  _D  F ) `
 x ) )
3029adantr 453 . . . . 5  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( *  o.  ( RR  _D  F ) ) `
 x )  =  ( ( RR  _D  F ) `  x
) )
318, 30eqtr3d 2318 . . . 4  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
* `  ( ( RR  _D  F ) `  x ) )  =  ( ( RR  _D  F ) `  x
) )
325, 31cjrebd 11681 . . 3  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  RR )
3332ralrimiva 2627 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  A. x  e.  dom  ( RR  _D  F
) ( ( RR 
_D  F ) `  x )  e.  RR )
34 ffnfv 5646 . 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 647 1  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544    C_ wss 3153    e. cmpt 4078   dom cdm 4688    o. ccom 4692    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   *ccj 11575    _D cdv 19207
This theorem is referenced by:  dvnfre  19295  dvferm1lem  19325  dvferm1  19326  dvferm2lem  19327  dvferm2  19328  dvferm  19329  c1lip2  19339  dvle  19348  dvivthlem1  19349  dvivth  19351  dvne0  19352  dvfsumle  19362  dvfsumge  19363  dvmptrecl  19365
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-icc 10657  df-fz 10777  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-plusg 13215  df-mulr 13216  df-starv 13217  df-tset 13221  df-ple 13222  df-ds 13224  df-rest 13321  df-topn 13322  df-topgen 13338  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-cncf 18376  df-limc 19210  df-dv 19211
  Copyright terms: Public domain W3C validator