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Theorem dvfre 19705
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 19662 . . 3  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
2 ffn 5532 . . 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 5809 . . . . 5  |-  ( x  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  x
)  e.  CC )
54adantl 453 . . . 4  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  CC )
6 simpr 448 . . . . . 6  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
7 fvco3 5740 . . . . . 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 645 . . . . 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 8981 . . . . . . . . . 10  |-  RR  C_  CC
10 fss 5540 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
119, 10mpan2 653 . . . . . . . . 9  |-  ( F : A --> RR  ->  F : A --> CC )
12 dvcj 19704 . . . . . . . . 9  |-  ( ( F : A --> CC  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
1311, 12sylan 458 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
14 ffvelrn 5808 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  y  e.  A )  ->  ( F `  y
)  e.  RR )
1514adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  RR )
1615cjred 11959 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( * `  ( F `  y
) )  =  ( F `  y ) )
1716mpteq2dva 4237 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( y  e.  A  |->  ( * `  ( F `  y )
) )  =  ( y  e.  A  |->  ( F `  y ) ) )
1815recnd 9048 . . . . . . . . . . 11  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  CC )
19 simpl 444 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F : A --> RR )
2019feqmptd 5719 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
21 cjf 11837 . . . . . . . . . . . . 13  |-  * : CC --> CC
2221a1i 11 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  * : CC --> CC )
2322feqmptd 5719 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  *  =  ( z  e.  CC  |->  ( * `  z ) ) )
24 fveq2 5669 . . . . . . . . . . 11  |-  ( z  =  ( F `  y )  ->  (
* `  z )  =  ( * `  ( F `  y ) ) )
2518, 20, 23, 24fmptco 5841 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  F
)  =  ( y  e.  A  |->  ( * `
 ( F `  y ) ) ) )
2617, 25, 203eqtr4d 2430 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  F
)  =  F )
2726oveq2d 6037 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( RR 
_D  F ) )
2813, 27eqtr3d 2422 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( RR  _D  F ) )
2928fveq1d 5671 . . . . . 6  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( ( *  o.  ( RR  _D  F
) ) `  x
)  =  ( ( RR  _D  F ) `
 x ) )
3029adantr 452 . . . . 5  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( *  o.  ( RR  _D  F ) ) `
 x )  =  ( ( RR  _D  F ) `  x
) )
318, 30eqtr3d 2422 . . . 4  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
* `  ( ( RR  _D  F ) `  x ) )  =  ( ( RR  _D  F ) `  x
) )
325, 31cjrebd 11935 . . 3  |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  RR )
3332ralrimiva 2733 . 2  |-  ( ( F : A --> RR  /\  A  C_  RR )  ->  A. x  e.  dom  ( RR  _D  F
) ( ( RR 
_D  F ) `  x )  e.  RR )
34 ffnfv 5834 . 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 646 1  |-  ( ( F : A --> RR  /\  A  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650    C_ wss 3264    e. cmpt 4208   dom cdm 4819    o. ccom 4823    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   *ccj 11829    _D cdv 19618
This theorem is referenced by:  dvnfre  19706  dvferm1lem  19736  dvferm1  19737  dvferm2lem  19738  dvferm2  19739  dvferm  19740  c1lip2  19750  dvle  19759  dvivthlem1  19760  dvivth  19762  dvne0  19763  dvfsumle  19773  dvfsumge  19774  dvmptrecl  19776
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-icc 10856  df-fz 10977  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-plusg 13470  df-mulr 13471  df-starv 13472  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-rest 13578  df-topn 13579  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-cncf 18780  df-limc 19621  df-dv 19622
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