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Theorem dvcj 19261
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 ) ) )

Proof of Theorem dvcj
StepHypRef Expression
1 dvf 19219 . . . . 5  |-  ( RR 
_D  ( *  o.  F ) ) : dom  ( RR  _D  ( *  o.  F
) ) --> CC
2 ffun 5329 . . . . 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 733 . . . . 5  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  F : X --> CC )
5 simplr 734 . . . . 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 19260 . . . 4  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  F
) )  ->  x
( RR  _D  (
*  o.  F ) ) ( * `  ( ( RR  _D  F ) `  x
) ) )
8 funbrfv 5495 . . . 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 4080 . 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 11554 . . . . . . . . . . . . 13  |-  * : CC --> CC
12 fco 5336 . . . . . . . . . . . . 13  |-  ( ( * : CC --> CC  /\  F : X --> CC )  ->  ( *  o.  F ) : X --> CC )
1311, 12mpan 654 . . . . . . . . . . . 12  |-  ( F : X --> CC  ->  ( *  o.  F ) : X --> CC )
1413ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  dom  ( RR  _D  (
*  o.  F ) ) )  ->  (
*  o.  F ) : X --> CC )
15 simplr 734 . . . . . . . . . . 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 19260 . . . . . . . . . 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 2766 . . . . . . . . . . 11  |-  x  e. 
_V
19 fvex 5472 . . . . . . . . . . 11  |-  ( * `
 ( ( RR 
_D  ( *  o.  F ) ) `  x ) )  e. 
_V
2018, 19breldm 4871 . . . . . . . . . 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 3160 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  ( *  o.  ( *  o.  F
) ) ) )
24 ffvelrn 5597 . . . . . . . . . . . . 13  |-  ( ( F : X --> CC  /\  x  e.  X )  ->  ( F `  x
)  e.  CC )
2524adantlr 698 . . . . . . . . . . . 12  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( F `  x )  e.  CC )
2625cjcjd 11649 . . . . . . . . . . 11  |-  ( ( ( F : X --> CC  /\  X  C_  RR )  /\  x  e.  X
)  ->  ( * `  ( * `  ( F `  x )
) )  =  ( F `  x ) )
2726mpteq2dva 4080 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( x  e.  X  |->  ( * `  (
* `  ( F `  x ) ) ) )  =  ( x  e.  X  |->  ( F `
 x ) ) )
2825cjcld 11646 . . . . . . . . . . 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 5509 . . . . . . . . . . . 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 5509 . . . . . . . . . . . 12  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  *  =  ( y  e.  CC  |->  ( * `  y ) ) )
33 fveq2 5458 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
* `  y )  =  ( * `  ( F `  x ) ) )
3425, 30, 32, 33fmptco 5625 . . . . . . . . . . 11  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  F
)  =  ( x  e.  X  |->  ( * `
 ( F `  x ) ) ) )
35 fveq2 5458 . . . . . . . . . . 11  |-  ( y  =  ( * `  ( F `  x ) )  ->  ( * `  y )  =  ( * `  ( * `
 ( F `  x ) ) ) )
3628, 34, 32, 35fmptco 5625 . . . . . . . . . 10  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  ( x  e.  X  |->  ( * `
 ( * `  ( F `  x ) ) ) ) )
3727, 36, 303eqtr4d 2300 . . . . . . . . 9  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  (
*  o.  F ) )  =  F )
3837oveq2d 5808 . . . . . . . 8  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  ( RR 
_D  F ) )
3938dmeqd 4869 . . . . . . 7  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  ( *  o.  F ) ) )  =  dom  ( RR  _D  F ) )
4023, 39sseqtrd 3189 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  C_  dom  ( RR 
_D  F ) )
41 fvex 5472 . . . . . . . . . 10  |-  ( * `
 ( ( RR 
_D  F ) `  x ) )  e. 
_V
4218, 41breldm 4871 . . . . . . . . 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 3160 . . . . . 6  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  F
)  C_  dom  ( RR 
_D  ( *  o.  F ) ) )
4640, 45eqssd 3171 . . . . 5  |-  ( ( F : X --> CC  /\  X  C_  RR )  ->  dom  ( RR  _D  (
*  o.  F ) )  =  dom  ( RR  _D  F ) )
4746feq2d 5318 . . . 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 5509 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  (
*  o.  F ) )  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  (
*  o.  F ) ) `  x ) ) )
50 dvf 19219 . . . . 5  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
5150ffvelrni 5598 . . . 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 5509 . . 3  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( RR  _D  F
)  =  ( x  e.  dom  ( RR 
_D  F )  |->  ( ( RR  _D  F
) `  x )
) )
55 fveq2 5458 . . 3  |-  ( y  =  ( ( RR 
_D  F ) `  x )  ->  (
* `  y )  =  ( * `  ( ( RR  _D  F ) `  x
) ) )
5652, 54, 32, 55fmptco 5625 . 2  |-  ( ( F : X --> CC  /\  X  C_  RR )  -> 
( *  o.  ( RR  _D  F ) )  =  ( x  e. 
dom  ( RR  _D  F )  |->  ( * `
 ( ( RR 
_D  F ) `  x ) ) ) )
5710, 49, 563eqtr4d 2300 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 1619    e. wcel 1621    C_ wss 3127   class class class wbr 3997    e. cmpt 4051   dom cdm 4661    o. ccom 4665   Fun wfun 4667   -->wf 4669   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   *ccj 11546    _D cdv 19175
This theorem is referenced by:  dvfre  19262  dvmptcj  19279
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-xneg 10419  df-xadd 10420  df-xmul 10421  df-ioo 10626  df-icc 10629  df-fz 10749  df-seq 11013  df-exp 11071  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-plusg 13183  df-mulr 13184  df-starv 13185  df-tset 13189  df-ple 13190  df-ds 13192  df-rest 13289  df-topn 13290  df-topgen 13306  df-xmet 16335  df-met 16336  df-bl 16337  df-mopn 16338  df-cnfld 16340  df-top 16598  df-bases 16600  df-topon 16601  df-topsp 16602  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-lp 16830  df-perf 16831  df-cn 16919  df-cnp 16920  df-haus 17005  df-fbas 17482  df-fg 17483  df-fil 17503  df-fm 17595  df-flim 17596  df-flf 17597  df-xms 17847  df-ms 17848  df-cncf 18344  df-limc 19178  df-dv 19179
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