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Theorem dvcj 22085
Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 22084. (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 dvf 22043 . . . . 5  |-  ( RR 
_D  ( *  o.  F ) ) : dom  ( RR  _D  ( *  o.  F
) ) --> CC
2 ffun 5731 . . . . 5  |-  ( ( RR  _D  ( *  o.  F ) ) : dom  ( RR 
_D  ( *  o.  F ) ) --> CC 
->  Fun  ( RR  _D  ( *  o.  F
) ) )
31, 2ax-mp 5 . . . 4  |-  Fun  ( RR  _D  ( *  o.  F ) )
4 simpll 753 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  F : X --> CC )
5 simplr 754 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  X  C_  RR )
6 simpr 461 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
74, 5, 6dvcjbr 22084 . . . 4  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) ) )
8 funbrfv 5904 . . . 4  |-  ( Fun  ( RR  _D  (
*  o.  F ) )  ->  ( x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `  x
) ) ) )
93, 7, 8mpsyl 63 . . 3  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
109mpteq2dva 4533 . 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
) ) ) )
11 cjf 12894 . . . . . . . . . . . . 13  |-  * : CC --> CC
12 fco 5739 . . . . . . . . . . . . 13  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
1311, 12mpan 670 . . . . . . . . . . . 12  |-  ( F : X --> CC  ->  ( *  o.  F ) : X --> CC )
1413ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
*  o.  F ) : X --> CC )
15 simplr 754 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  X  C_  RR )
16 simpr 461 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
1714, 15, 16dvcjbr 22084 . . . . . . . . . 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
) ) )
18 vex 3116 . . . . . . . . . . 11  |-  x  e. 
_V
19 fvex 5874 . . . . . . . . . . 11  |-  ( * `
 ( ( RR 
_D  ( *  o.  F ) ) `  x ) )  e. 
_V
2018, 19breldm 5205 . . . . . . . . . 10  |-  ( x ( RR  _D  (
*  o.  ( *  o.  F ) ) ) ( * `  ( ( RR  _D  ( *  o.  F
) ) `  x
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  (
*  o.  F ) ) ) )
2117, 20syl 16 . . . . . . . . 9  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  (
*  o.  F ) ) ) )
2221ex 434 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  (
*  o.  F ) )  ->  x  e.  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) ) ) )
2322ssrdv 3510 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  ( *  o.  ( *  o.  F
) ) ) )
24 ffvelrn 6017 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
2524adantlr 714 . . . . . . . . . . . 12  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( F `  x )  e.  CC )
2625cjcjd 12989 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( * `  ( F `  x )
) )  =  ( F `  x ) )
2726mpteq2dva 4533 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  X  |->  ( * `  (
* `  ( F `  x ) ) ) )  =  ( x  e.  X  |->  ( F `
 x ) ) )
2825cjcld 12986 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( F `  x
) )  e.  CC )
29 simpl 457 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
3029feqmptd 5918 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
3111a1i 11 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  * : CC --> CC )
3231feqmptd 5918 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  *  =  ( y  e.  CC  |->  ( * `  y ) ) )
33 fveq2 5864 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
* `  y )  =  ( * `  ( F `  x ) ) )
3425, 30, 32, 33fmptco 6052 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
)  =  ( x  e.  X  |->  ( * `
 ( F `  x ) ) ) )
35 fveq2 5864 . . . . . . . . . . 11  |-  ( y  =  ( * `  ( F `  x ) )  ->  ( * `  y )  =  ( * `  ( * `
 ( F `  x ) ) ) )
3628, 34, 32, 35fmptco 6052 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  ( x  e.  X  |->  ( * `
 ( * `  ( F `  x ) ) ) ) )
3727, 36, 303eqtr4d 2518 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  F )
3837oveq2d 6298 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  ( RR 
_D  F ) )
3938dmeqd 5203 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  dom  ( RR  _D  F ) )
4023, 39sseqtrd 3540 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  F ) )
41 fvex 5874 . . . . . . . . . 10  |-  ( * `
 ( ( RR 
_D  F ) `  x ) )  e. 
_V
4218, 41breldm 5205 . . . . . . . . 9  |-  ( x ( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
437, 42syl 16 . . . . . . . 8  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
4443ex 434 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  F
)  ->  x  e.  dom  ( RR  _D  (
*  o.  F ) ) ) )
4544ssrdv 3510 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  dom  ( RR 
_D  ( *  o.  F ) ) )
4640, 45eqssd 3521 . . . . 5  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  =  dom  ( RR  _D  F ) )
4746feq2d 5716 . . . 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 ) )
481, 47mpbii 211 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) ) : dom  ( RR  _D  F ) --> CC )
4948feqmptd 5918 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  (
*  o.  F ) ) `  x ) ) )
50 dvf 22043 . . . . 5  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
5150ffvelrni 6018 . . . 4  |-  ( x  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  x
)  e.  CC )
5251adantl 466 . . 3  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  CC )
5350a1i 11 . . . 4  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> CC )
5453feqmptd 5918 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
)  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  F
) `  x )
) )
55 fveq2 5864 . . 3  |-  ( y  =  ( ( RR 
_D  F ) `  x )  ->  (
* `  y )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
5652, 54, 32, 55fmptco 6052 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( x  e. 
dom  ( RR  _D  F )  |->  ( * `
 ( ( RR 
_D  F ) `  x ) ) ) )
5710, 49, 563eqtr4d 2518 1  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999    o. ccom 5003   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   *ccj 12886    _D cdv 21999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-seq 12071  df-exp 12130  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-plusg 14561  df-mulr 14562  df-starv 14563  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-rest 14671  df-topn 14672  df-topgen 14692  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-cncf 21114  df-limc 22002  df-dv 22003
This theorem is referenced by:  dvfre  22086  dvmptcj  22103
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