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Theorem dvcj 22781
Description: The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 22780. (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 22739 . . . . 5  |-  ( RR 
_D  ( *  o.  F ) ) : dom  ( RR  _D  ( *  o.  F
) ) --> CC
2 ffun 5748 . . . . 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 758 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  F : X --> CC )
5 simplr 760 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  X  C_  RR )
6 simpr 462 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
74, 5, 6dvcjbr 22780 . . . 4  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) ) )
8 funbrfv 5919 . . . 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 65 . . 3  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
109mpteq2dva 4512 . 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 13146 . . . . . . . . . . . . 13  |-  * : CC --> CC
12 fco 5756 . . . . . . . . . . . . 13  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
1311, 12mpan 674 . . . . . . . . . . . 12  |-  ( F : X --> CC  ->  ( *  o.  F ) : X --> CC )
1413ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
*  o.  F ) : X --> CC )
15 simplr 760 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  X  C_  RR )
16 simpr 462 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
1714, 15, 16dvcjbr 22780 . . . . . . . . . 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 3090 . . . . . . . . . . 11  |-  x  e. 
_V
19 fvex 5891 . . . . . . . . . . 11  |-  ( * `
 ( ( RR 
_D  ( *  o.  F ) ) `  x ) )  e. 
_V
2018, 19breldm 5059 . . . . . . . . . 10  |-  ( x ( RR  _D  (
*  o.  ( *  o.  F ) ) ) ( * `  ( ( RR  _D  ( *  o.  F
) ) `  x
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  (
*  o.  F ) ) ) )
2117, 20syl 17 . . . . . . . . 9  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  (
*  o.  F ) ) ) )
2221ex 435 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  (
*  o.  F ) )  ->  x  e.  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) ) ) )
2322ssrdv 3476 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  ( *  o.  ( *  o.  F
) ) ) )
24 ffvelrn 6035 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
2524adantlr 719 . . . . . . . . . . . 12  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( F `  x )  e.  CC )
2625cjcjd 13241 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( * `  ( F `  x )
) )  =  ( F `  x ) )
2726mpteq2dva 4512 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  X  |->  ( * `  (
* `  ( F `  x ) ) ) )  =  ( x  e.  X  |->  ( F `
 x ) ) )
2825cjcld 13238 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( F `  x
) )  e.  CC )
29 simpl 458 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
3029feqmptd 5934 . . . . . . . . . . . 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 5934 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  *  =  ( y  e.  CC  |->  ( * `  y ) ) )
33 fveq2 5881 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
* `  y )  =  ( * `  ( F `  x ) ) )
3425, 30, 32, 33fmptco 6071 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
)  =  ( x  e.  X  |->  ( * `
 ( F `  x ) ) ) )
35 fveq2 5881 . . . . . . . . . . 11  |-  ( y  =  ( * `  ( F `  x ) )  ->  ( * `  y )  =  ( * `  ( * `
 ( F `  x ) ) ) )
3628, 34, 32, 35fmptco 6071 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  ( x  e.  X  |->  ( * `
 ( * `  ( F `  x ) ) ) ) )
3727, 36, 303eqtr4d 2480 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  F )
3837oveq2d 6321 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  ( RR 
_D  F ) )
3938dmeqd 5057 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  dom  ( RR  _D  F ) )
4023, 39sseqtrd 3506 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  F ) )
41 fvex 5891 . . . . . . . . . 10  |-  ( * `
 ( ( RR 
_D  F ) `  x ) )  e. 
_V
4218, 41breldm 5059 . . . . . . . . 9  |-  ( x ( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
437, 42syl 17 . . . . . . . 8  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
4443ex 435 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  F
)  ->  x  e.  dom  ( RR  _D  (
*  o.  F ) ) ) )
4544ssrdv 3476 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  dom  ( RR 
_D  ( *  o.  F ) ) )
4640, 45eqssd 3487 . . . . 5  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  =  dom  ( RR  _D  F ) )
4746feq2d 5733 . . . 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 214 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) ) : dom  ( RR  _D  F ) --> CC )
4948feqmptd 5934 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  (
*  o.  F ) ) `  x ) ) )
50 dvf 22739 . . . . 5  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
5150ffvelrni 6036 . . . 4  |-  ( x  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  x
)  e.  CC )
5251adantl 467 . . 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 5934 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
)  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  F
) `  x )
) )
55 fveq2 5881 . . 3  |-  ( y  =  ( ( RR 
_D  F ) `  x )  ->  (
* `  y )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
5652, 54, 32, 55fmptco 6071 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( x  e. 
dom  ( RR  _D  F )  |->  ( * `
 ( ( RR 
_D  F ) `  x ) ) ) )
5710, 49, 563eqtr4d 2480 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 370    = wceq 1437    e. wcel 1870    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854    o. ccom 4858   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   *ccj 13138    _D cdv 22695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-rest 15280  df-topn 15281  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-cncf 21806  df-limc 22698  df-dv 22699
This theorem is referenced by:  dvfre  22782  dvmptcj  22799
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