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Theorem dvcj 12831
Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 12830. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
Assertion
Ref Expression
dvcj  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )

Proof of Theorem dvcj
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 518 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  F : X --> CC )
2 simplr 519 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  X  C_  RR )
3 simpr 109 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
41, 2, 3dvcjbr 12830 . . . 4  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) ) )
5 cjf 10612 . . . . . . . . . . . 12  |-  * : CC --> CC
6 fco 5283 . . . . . . . . . . . 12  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
75, 6mpan 420 . . . . . . . . . . 11  |-  ( F : X --> CC  ->  ( *  o.  F ) : X --> CC )
87adantr 274 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
) : X --> CC )
97fdmd 5274 . . . . . . . . . . . 12  |-  ( F : X --> CC  ->  dom  ( *  o.  F
)  =  X )
109adantr 274 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( *  o.  F
)  =  X )
1110feq2d 5255 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( ( *  o.  F ) : dom  ( *  o.  F
) --> CC  <->  ( *  o.  F ) : X --> CC ) )
128, 11mpbird 166 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
) : dom  (
*  o.  F ) --> CC )
13 simpr 109 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  X  C_  RR )
1410, 13eqsstrd 3128 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( *  o.  F
)  C_  RR )
15 cnex 7737 . . . . . . . . . 10  |-  CC  e.  _V
16 reex 7747 . . . . . . . . . 10  |-  RR  e.  _V
1715, 16elpm2 6567 . . . . . . . . 9  |-  ( ( *  o.  F )  e.  ( CC  ^pm  RR )  <->  ( ( *  o.  F ) : dom  ( *  o.  F ) --> CC  /\  dom  ( *  o.  F
)  C_  RR )
)
1812, 14, 17sylanbrc 413 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
)  e.  ( CC 
^pm  RR ) )
19 dvfpm 12816 . . . . . . . 8  |-  ( ( *  o.  F )  e.  ( CC  ^pm  RR )  ->  ( RR  _D  ( *  o.  F
) ) : dom  ( RR  _D  (
*  o.  F ) ) --> CC )
2018, 19syl 14 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) ) : dom  ( RR  _D  ( *  o.  F ) ) --> CC )
2120ffund 5271 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  Fun  ( RR  _D  (
*  o.  F ) ) )
22 funbrfv 5453 . . . . . 6  |-  ( Fun  ( RR  _D  (
*  o.  F ) )  ->  ( x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `  x
) ) ) )
2321, 22syl 14 . . . . 5  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x ( RR 
_D  ( *  o.  F ) ) ( * `  ( ( RR  _D  F ) `
 x ) )  ->  ( ( RR 
_D  ( *  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `
 x ) ) ) )
2423adantr 274 . . . 4  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
x ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  x ) )  -> 
( ( RR  _D  ( *  o.  F
) ) `  x
)  =  ( * `
 ( ( RR 
_D  F ) `  x ) ) ) )
254, 24mpd 13 . . 3  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
2625mpteq2dva 4013 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  F
)  |->  ( ( RR 
_D  ( *  o.  F ) ) `  x ) )  =  ( x  e.  dom  ( RR  _D  F
)  |->  ( * `  ( ( RR  _D  F ) `  x
) ) ) )
27 vex 2684 . . . . . . . . . 10  |-  x  e. 
_V
2820ffvelrnda 5548 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  e.  CC )
2928cjcld 10705 . . . . . . . . . 10  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
* `  ( ( RR  _D  ( *  o.  F ) ) `  x ) )  e.  CC )
307ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
*  o.  F ) : X --> CC )
31 simplr 519 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  X  C_  RR )
32 simpr 109 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
3330, 31, 32dvcjbr 12830 . . . . . . . . . 10  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x
( RR  _D  (
*  o.  ( *  o.  F ) ) ) ( * `  ( ( RR  _D  ( *  o.  F
) ) `  x
) ) )
34 breldmg 4740 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  ( * `  (
( RR  _D  (
*  o.  F ) ) `  x ) )  e.  CC  /\  x ( RR  _D  ( *  o.  (
*  o.  F ) ) ) ( * `
 ( ( RR 
_D  ( *  o.  F ) ) `  x ) ) )  ->  x  e.  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) ) )
3527, 29, 33, 34mp3an2i 1320 . . . . . . . . 9  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  (
*  o.  F ) ) ) )
3635ex 114 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  (
*  o.  F ) )  ->  x  e.  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) ) ) )
3736ssrdv 3098 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  ( *  o.  ( *  o.  F
) ) ) )
38 ffvelrn 5546 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
3938adantlr 468 . . . . . . . . . . . 12  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( F `  x )  e.  CC )
4039cjcjd 10708 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( * `  ( F `  x )
) )  =  ( F `  x ) )
4140mpteq2dva 4013 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  X  |->  ( * `  (
* `  ( F `  x ) ) ) )  =  ( x  e.  X  |->  ( F `
 x ) ) )
4239cjcld 10705 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( F `  x
) )  e.  CC )
43 simpl 108 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
4443feqmptd 5467 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
455a1i 9 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  * : CC --> CC )
4645feqmptd 5467 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  *  =  ( y  e.  CC  |->  ( * `  y ) ) )
47 fveq2 5414 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
* `  y )  =  ( * `  ( F `  x ) ) )
4839, 44, 46, 47fmptco 5579 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
)  =  ( x  e.  X  |->  ( * `
 ( F `  x ) ) ) )
49 fveq2 5414 . . . . . . . . . . 11  |-  ( y  =  ( * `  ( F `  x ) )  ->  ( * `  y )  =  ( * `  ( * `
 ( F `  x ) ) ) )
5042, 48, 46, 49fmptco 5579 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  ( x  e.  X  |->  ( * `
 ( * `  ( F `  x ) ) ) ) )
5141, 50, 443eqtr4d 2180 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  F )
5251oveq2d 5783 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  ( RR 
_D  F ) )
5352dmeqd 4736 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  dom  ( RR  _D  F ) )
5437, 53sseqtrd 3130 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  F ) )
55 ffdm 5288 . . . . . . . . . . . . 13  |-  ( F : X --> CC  ->  ( F : dom  F --> CC  /\  dom  F  C_  X ) )
5655simpld 111 . . . . . . . . . . . 12  |-  ( F : X --> CC  ->  F : dom  F --> CC )
5756adantr 274 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : dom  F --> CC )
58 fdm 5273 . . . . . . . . . . . . 13  |-  ( F : X --> CC  ->  dom 
F  =  X )
5958adantr 274 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  F  =  X )
6059, 13eqsstrd 3128 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  F  C_  RR )
6115, 16elpm2 6567 . . . . . . . . . . 11  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
6257, 60, 61sylanbrc 413 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
63 dvfpm 12816 . . . . . . . . . 10  |-  ( F  e.  ( CC  ^pm  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC )
6462, 63syl 14 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> CC )
6564ffvelrnda 5548 . . . . . . . 8  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  CC )
6665cjcld 10705 . . . . . . 7  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
* `  ( ( RR  _D  F ) `  x ) )  e.  CC )
67 breldmg 4740 . . . . . . 7  |-  ( ( x  e.  _V  /\  ( * `  (
( RR  _D  F
) `  x )
)  e.  CC  /\  x ( RR  _D  ( *  o.  F
) ) ( * `
 ( ( RR 
_D  F ) `  x ) ) )  ->  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )
6827, 66, 4, 67mp3an2i 1320 . . . . . 6  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
6954, 68eqelssd 3111 . . . . 5  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  =  dom  ( RR  _D  F ) )
7069feq2d 5255 . . . 4  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( ( RR  _D  ( *  o.  F
) ) : dom  ( RR  _D  (
*  o.  F ) ) --> CC  <->  ( RR  _D  ( *  o.  F
) ) : dom  ( RR  _D  F
) --> CC ) )
7120, 70mpbid 146 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) ) : dom  ( RR  _D  F ) --> CC )
7271feqmptd 5467 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  (
*  o.  F ) ) `  x ) ) )
7364feqmptd 5467 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
)  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  F
) `  x )
) )
74 fveq2 5414 . . 3  |-  ( y  =  ( ( RR 
_D  F ) `  x )  ->  (
* `  y )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
7565, 73, 46, 74fmptco 5579 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( x  e. 
dom  ( RR  _D  F )  |->  ( * `
 ( ( RR 
_D  F ) `  x ) ) ) )
7626, 72, 753eqtr4d 2180 1  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2681    C_ wss 3066   class class class wbr 3924    |-> cmpt 3984   dom cdm 4534    o. ccom 4538   Fun wfun 5112   -->wf 5114   ` cfv 5118  (class class class)co 5767    ^pm cpm 6536   CCcc 7611   RRcr 7612   *ccj 10604    _D cdv 12782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-map 6537  df-pm 6538  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-xneg 9552  df-xadd 9553  df-ioo 9668  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-rest 12111  df-topgen 12130  df-psmet 12145  df-xmet 12146  df-met 12147  df-bl 12148  df-mopn 12149  df-top 12154  df-topon 12167  df-bases 12199  df-ntr 12254  df-cn 12346  df-cnp 12347  df-cncf 12716  df-limced 12783  df-dvap 12784
This theorem is referenced by:  dvfre  12832
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