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Theorem dvcncxp1 28477
Description: Derivative of complex power with respect to first argument on the complex plane. (Contributed by Brendan Leahy, 18-Dec-2018.)
Hypothesis
Ref Expression
dvcncxp1.d  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
Assertion
Ref Expression
dvcncxp1  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  D  |->  ( x  ^c  A ) ) )  =  ( x  e.  D  |->  ( A  x.  ( x  ^c 
( A  -  1 ) ) ) ) )
Distinct variable groups:    x, A    x, D

Proof of Theorem dvcncxp1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnelprrecn 9375 . . . 4  |-  CC  e.  { RR ,  CC }
21a1i 11 . . 3  |-  ( A  e.  CC  ->  CC  e.  { RR ,  CC } )
3 dvcncxp1.d . . . . . . 7  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
4 difss 3483 . . . . . . 7  |-  ( CC 
\  ( -oo (,] 0 ) )  C_  CC
53, 4eqsstri 3386 . . . . . 6  |-  D  C_  CC
65sseli 3352 . . . . 5  |-  ( x  e.  D  ->  x  e.  CC )
73logdmn0 22085 . . . . 5  |-  ( x  e.  D  ->  x  =/=  0 )
86, 7logcld 22022 . . . 4  |-  ( x  e.  D  ->  ( log `  x )  e.  CC )
98adantl 466 . . 3  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( log `  x
)  e.  CC )
106, 7reccld 10100 . . . 4  |-  ( x  e.  D  ->  (
1  /  x )  e.  CC )
1110adantl 466 . . 3  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( 1  /  x
)  e.  CC )
12 mulcl 9366 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
13 efcl 13368 . . . 4  |-  ( ( A  x.  y )  e.  CC  ->  ( exp `  ( A  x.  y ) )  e.  CC )
1412, 13syl 16 . . 3  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
15 ovex 6116 . . . 4  |-  ( ( exp `  ( A  x.  y ) )  x.  A )  e. 
_V
1615a1i 11 . . 3  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
173dvlog 22096 . . . 4  |-  ( CC 
_D  ( log  |`  D ) )  =  ( x  e.  D  |->  ( 1  /  x ) )
183logcn 22092 . . . . . . . 8  |-  ( log  |`  D )  e.  ( D -cn-> CC )
19 cncff 20469 . . . . . . . 8  |-  ( ( log  |`  D )  e.  ( D -cn-> CC )  ->  ( log  |`  D ) : D --> CC )
2018, 19mp1i 12 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  D ) : D --> CC )
2120feqmptd 5744 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  D )  =  ( x  e.  D  |->  ( ( log  |`  D ) `
 x ) ) )
22 fvres 5704 . . . . . . 7  |-  ( x  e.  D  ->  (
( log  |`  D ) `
 x )  =  ( log `  x
) )
2322mpteq2ia 4374 . . . . . 6  |-  ( x  e.  D  |->  ( ( log  |`  D ) `  x ) )  =  ( x  e.  D  |->  ( log `  x
) )
2421, 23syl6eq 2491 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  D )  =  ( x  e.  D  |->  ( log `  x
) ) )
2524oveq2d 6107 . . . 4  |-  ( A  e.  CC  ->  ( CC  _D  ( log  |`  D ) )  =  ( CC 
_D  ( x  e.  D  |->  ( log `  x
) ) ) )
2617, 25syl5reqr 2490 . . 3  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  D  |->  ( log `  x
) ) )  =  ( x  e.  D  |->  ( 1  /  x
) ) )
27 simpl 457 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  A  e.  CC )
28 efcl 13368 . . . . 5  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
2928adantl 466 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  x
)  e.  CC )
30 simpr 461 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  y  e.  CC )
31 1cnd 9402 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
322dvmptid 21431 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  y ) )  =  ( y  e.  CC  |->  1 ) )
33 id 22 . . . . . 6  |-  ( A  e.  CC  ->  A  e.  CC )
342, 30, 31, 32, 33dvmptcmul 21438 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
35 mulid1 9383 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
3635mpteq2dv 4379 . . . . 5  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
3734, 36eqtrd 2475 . . . 4  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
38 dvef 21452 . . . . 5  |-  ( CC 
_D  exp )  =  exp
39 eff 13367 . . . . . . . 8  |-  exp : CC
--> CC
4039a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  exp : CC --> CC )
4140feqmptd 5744 . . . . . 6  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
4241oveq2d 6107 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  exp )  =  ( CC  _D  (
x  e.  CC  |->  ( exp `  x ) ) ) )
4338, 42, 413eqtr3a 2499 . . . 4  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
44 fveq2 5691 . . . 4  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
452, 2, 12, 27, 29, 29, 37, 43, 44, 44dvmptco 21446 . . 3  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  CC  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
46 oveq2 6099 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
4746fveq2d 5695 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
4847oveq1d 6106 . . 3  |-  ( y  =  ( log `  x
)  ->  ( ( exp `  ( A  x.  y ) )  x.  A )  =  ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A ) )
492, 2, 9, 11, 14, 16, 26, 45, 47, 48dvmptco 21446 . 2  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  D  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )  =  ( x  e.  D  |->  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) ) )
506adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  x  e.  CC )
517adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  x  =/=  0 )
52 simpl 457 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  A  e.  CC )
5350, 51, 52cxpefd 22157 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( x  ^c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
5453mpteq2dva 4378 . . 3  |-  ( A  e.  CC  ->  (
x  e.  D  |->  ( x  ^c  A ) )  =  ( x  e.  D  |->  ( exp `  ( A  x.  ( log `  x
) ) ) ) )
5554oveq2d 6107 . 2  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  D  |->  ( x  ^c  A ) ) )  =  ( CC  _D  ( x  e.  D  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) ) )
56 1cnd 9402 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  1  e.  CC )
5750, 51, 52, 56cxpsubd 22163 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( x  ^c 
( A  -  1 ) )  =  ( ( x  ^c  A )  /  (
x  ^c  1 ) ) )
5850cxp1d 22151 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( x  ^c 
1 )  =  x )
5958oveq2d 6107 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( ( x  ^c  A )  /  (
x  ^c  1 ) )  =  ( ( x  ^c  A )  /  x
) )
6050, 52cxpcld 22153 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( x  ^c  A )  e.  CC )
6160, 50, 51divrecd 10110 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( ( x  ^c  A )  /  x
)  =  ( ( x  ^c  A )  x.  ( 1  /  x ) ) )
6257, 59, 613eqtrd 2479 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( x  ^c 
( A  -  1 ) )  =  ( ( x  ^c  A )  x.  (
1  /  x ) ) )
6362oveq2d 6107 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( A  x.  (
x  ^c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^c  A )  x.  ( 1  /  x ) ) ) )
6452, 60, 11mul12d 9578 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( A  x.  (
( x  ^c  A )  x.  (
1  /  x ) ) )  =  ( ( x  ^c  A )  x.  ( A  x.  ( 1  /  x ) ) ) )
6560, 52, 11mulassd 9409 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( ( ( x  ^c  A )  x.  A )  x.  ( 1  /  x
) )  =  ( ( x  ^c  A )  x.  ( A  x.  ( 1  /  x ) ) ) )
6664, 65eqtr4d 2478 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( A  x.  (
( x  ^c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^c  A )  x.  A
)  x.  ( 1  /  x ) ) )
6753oveq1d 6106 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( ( x  ^c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
6867oveq1d 6106 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( ( ( x  ^c  A )  x.  A )  x.  ( 1  /  x
) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
6963, 66, 683eqtrd 2479 . . 3  |-  ( ( A  e.  CC  /\  x  e.  D )  ->  ( A  x.  (
x  ^c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
7069mpteq2dva 4378 . 2  |-  ( A  e.  CC  ->  (
x  e.  D  |->  ( A  x.  ( x  ^c  ( A  -  1 ) ) ) )  =  ( x  e.  D  |->  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) ) )
7149, 55, 703eqtr4d 2485 1  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  D  |->  ( x  ^c  A ) ) )  =  ( x  e.  D  |->  ( A  x.  ( x  ^c 
( A  -  1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972    \ cdif 3325   {cpr 3879    e. cmpt 4350    |` cres 4842   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    x. cmul 9287   -oocmnf 9416    - cmin 9595    / cdiv 9993   (,]cioc 11301   expce 13347   -cn->ccncf 20452    _D cdv 21338   logclog 22006    ^c ccxp 22007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-tan 13357  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-cmp 18990  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-cxp 22009
This theorem is referenced by:  dvcnsqr  28478
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