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

Theorem dvcj 19295
Description: The derivative of the conjugate of a function. (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 ) ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem dvcj
StepHypRef Expression
1 dvf 19253 . . . . 5  |-  ( RR 
_D  ( *  o.  F ) ) : dom  ( RR  _D  ( *  o.  F
) ) --> CC
2 ffun 5358 . . . . 5  |-  ( ( RR  _D  ( *  o.  F ) ) : dom  ( RR 
_D  ( *  o.  F ) ) --> CC 
->  Fun  ( RR  _D  ( *  o.  F
) ) )
31, 2ax-mp 10 . . . 4  |-  Fun  ( RR  _D  ( *  o.  F ) )
4 simpll 732 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  F : X --> CC )
5 simplr 733 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  X  C_  RR )
6 simpr 449 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x  e.  dom  ( RR  _D  F ) )
74, 5, 6dvcjbr 19294 . . . 4  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) ) )
8 funbrfv 5524 . . . 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 61 . . 3  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  (
*  o.  F ) ) `  x )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
109mpteq2dva 4109 . 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 11585 . . . . . . . . . . . . 13  |-  * : CC --> CC
12 fco 5365 . . . . . . . . . . . . 13  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
1311, 12mpan 653 . . . . . . . . . . . 12  |-  ( F : X --> CC  ->  ( *  o.  F ) : X --> CC )
1413ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
*  o.  F ) : X --> CC )
15 simplr 733 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  X  C_  RR )
16 simpr 449 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  x  e.  dom  ( RR  _D  ( *  o.  F
) ) )
1714, 15, 16dvcjbr 19294 . . . . . . . . . 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 2794 . . . . . . . . . . 11  |-  x  e. 
_V
19 fvex 5501 . . . . . . . . . . 11  |-  ( * `
 ( ( RR 
_D  ( *  o.  F ) ) `  x ) )  e. 
_V
2018, 19breldm 4884 . . . . . . . . . 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 425 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  (
*  o.  F ) )  ->  x  e.  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) ) ) )
2322ssrdv 3188 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  ( *  o.  ( *  o.  F
) ) ) )
24 ffvelrn 5626 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
2524adantlr 697 . . . . . . . . . . . 12  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( F `  x )  e.  CC )
2625cjcjd 11680 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( * `  ( F `  x )
) )  =  ( F `  x ) )
2726mpteq2dva 4109 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  X  |->  ( * `  (
* `  ( F `  x ) ) ) )  =  ( x  e.  X  |->  ( F `
 x ) ) )
2825cjcld 11677 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( F `  x
) )  e.  CC )
29 simpl 445 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F : X --> CC )
3029feqmptd 5538 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
3111a1i 12 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  * : CC --> CC )
3231feqmptd 5538 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  *  =  ( y  e.  CC  |->  ( * `  y ) ) )
33 fveq2 5487 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
* `  y )  =  ( * `  ( F `  x ) ) )
3425, 30, 32, 33fmptco 5654 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
)  =  ( x  e.  X  |->  ( * `
 ( F `  x ) ) ) )
35 fveq2 5487 . . . . . . . . . . 11  |-  ( y  =  ( * `  ( F `  x ) )  ->  ( * `  y )  =  ( * `  ( * `
 ( F `  x ) ) ) )
3628, 34, 32, 35fmptco 5654 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  ( x  e.  X  |->  ( * `
 ( * `  ( F `  x ) ) ) ) )
3727, 36, 303eqtr4d 2328 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  F )
3837oveq2d 5837 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  ( RR 
_D  F ) )
3938dmeqd 4882 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  dom  ( RR  _D  F ) )
4023, 39sseqtrd 3217 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  F ) )
41 fvex 5501 . . . . . . . . . 10  |-  ( * `
 ( ( RR 
_D  F ) `  x ) )  e. 
_V
4218, 41breldm 4884 . . . . . . . . 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 425 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  dom  ( RR  _D  F
)  ->  x  e.  dom  ( RR  _D  (
*  o.  F ) ) ) )
4544ssrdv 3188 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  dom  ( RR 
_D  ( *  o.  F ) ) )
4640, 45eqssd 3199 . . . . 5  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  =  dom  ( RR  _D  F ) )
4746feq2d 5347 . . . 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 204 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) ) : dom  ( RR  _D  F ) --> CC )
4948feqmptd 5538 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  (
*  o.  F ) ) `  x ) ) )
50 dvf 19253 . . . . 5  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
5150ffvelrni 5627 . . . 4  |-  ( x  e.  dom  ( RR 
_D  F )  -> 
( ( RR  _D  F ) `  x
)  e.  CC )
5251adantl 454 . . 3  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  x )  e.  CC )
5350a1i 12 . . . 4  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
) : dom  ( RR  _D  F ) --> CC )
5453feqmptd 5538 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
)  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  F
) `  x )
) )
55 fveq2 5487 . . 3  |-  ( y  =  ( ( RR 
_D  F ) `  x )  ->  (
* `  y )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
5652, 54, 32, 55fmptco 5654 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( x  e. 
dom  ( RR  _D  F )  |->  ( * `
 ( ( RR 
_D  F ) `  x ) ) ) )
5710, 49, 563eqtr4d 2328 1  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1625    e. wcel 1687    C_ wss 3155   class class class wbr 4026    e. cmpt 4080   dom cdm 4690    o. ccom 4694   Fun wfun 5217   -->wf 5219   ` cfv 5223  (class class class)co 5821   CCcc 8732   RRcr 8733   *ccj 11577    _D cdv 19209
This theorem is referenced by:  dvfre  19296  dvmptcj  19313
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812
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 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-oadd 6480  df-er 6657  df-map 6771  df-pm 6772  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  df-ioo 10656  df-icc 10659  df-fz 10779  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-plusg 13217  df-mulr 13218  df-starv 13219  df-tset 13223  df-ple 13224  df-ds 13226  df-rest 13323  df-topn 13324  df-topgen 13340  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-cncf 18378  df-limc 19212  df-dv 19213
  Copyright terms: Public domain W3C validator