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Theorem dvcxp1 22139
Description: The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)
Assertion
Ref Expression
dvcxp1  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  ^c  A ) ) )  =  ( x  e.  RR+  |->  ( A  x.  ( x  ^c 
( A  -  1 ) ) ) ) )
Distinct variable group:    x, A

Proof of Theorem dvcxp1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 reelprrecn 9370 . . . 4  |-  RR  e.  { RR ,  CC }
21a1i 11 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
3 relogcl 21986 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
43adantl 463 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
5 rpreccl 11010 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
65adantl 463 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
7 recn 9368 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 9362 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
9 efcl 13364 . . . . 5  |-  ( ( A  x.  y )  e.  CC  ->  ( exp `  ( A  x.  y ) )  e.  CC )
108, 9syl 16 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
117, 10sylan2 471 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
12 ovex 6115 . . . 4  |-  ( ( exp `  ( A  x.  y ) )  x.  A )  e. 
_V
1312a1i 11 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
14 dvrelog 22041 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
15 relogf1o 21977 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
16 f1of 5638 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1715, 16mp1i 12 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1817feqmptd 5741 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
19 fvres 5701 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2019mpteq2ia 4371 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2118, 20syl6eq 2489 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2221oveq2d 6106 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2314, 22syl5reqr 2488 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
24 eqid 2441 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2524cnfldtopon 20321 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
26 toponmax 18492 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2725, 26mp1i 12 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
28 ax-resscn 9335 . . . . . 6  |-  RR  C_  CC
2928a1i 11 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
30 df-ss 3339 . . . . 5  |-  ( RR  C_  CC  <->  ( RR  i^i  CC )  =  RR )
3129, 30sylib 196 . . . 4  |-  ( A  e.  CC  ->  ( RR  i^i  CC )  =  RR )
3212a1i 11 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
33 cnelprrecn 9371 . . . . . 6  |-  CC  e.  { RR ,  CC }
3433a1i 11 . . . . 5  |-  ( A  e.  CC  ->  CC  e.  { RR ,  CC } )
35 simpl 454 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  A  e.  CC )
36 efcl 13364 . . . . . 6  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3736adantl 463 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  x
)  e.  CC )
38 simpr 458 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  y  e.  CC )
39 1cnd 9398 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4034dvmptid 21390 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  y ) )  =  ( y  e.  CC  |->  1 ) )
41 id 22 . . . . . . 7  |-  ( A  e.  CC  ->  A  e.  CC )
4234, 38, 39, 40, 41dvmptcmul 21397 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
43 mulid1 9379 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4443mpteq2dv 4376 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4542, 44eqtrd 2473 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
46 eff 13363 . . . . . . . . . . 11  |-  exp : CC
--> CC
4746a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
4847feqmptd 5741 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
4948eqcomd 2446 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5049oveq2d 6106 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
51 dvef 21411 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5250, 51syl6eq 2489 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5352, 48eqtrd 2473 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
54 fveq2 5688 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5534, 34, 8, 35, 37, 37, 45, 53, 54, 54dvmptco 21405 . . . 4  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  CC  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
5624, 2, 27, 31, 10, 32, 55dvmptres3 21389 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
57 oveq2 6098 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
5857fveq2d 5692 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
5958oveq1d 6105 . . 3  |-  ( y  =  ( log `  x
)  ->  ( ( exp `  ( A  x.  y ) )  x.  A )  =  ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A ) )
602, 2, 4, 6, 11, 13, 23, 56, 58, 59dvmptco 21405 . 2  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )  =  ( x  e.  RR+  |->  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) ) )
61 rpcn 10995 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6261adantl 463 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
63 rpne0 11002 . . . . . 6  |-  ( x  e.  RR+  ->  x  =/=  0 )
6463adantl 463 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  =/=  0 )
65 simpl 454 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
6662, 64, 65cxpefd 22116 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6766mpteq2dva 4375 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
6867oveq2d 6106 . 2  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  ^c  A ) ) )  =  ( RR  _D  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) ) )
69 1cnd 9398 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
1  e.  CC )
7062, 64, 65, 69cxpsubd 22122 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c 
( A  -  1 ) )  =  ( ( x  ^c  A )  /  (
x  ^c  1 ) ) )
7162cxp1d 22110 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c 
1 )  =  x )
7271oveq2d 6106 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^c  A )  /  (
x  ^c  1 ) )  =  ( ( x  ^c  A )  /  x
) )
7362, 65cxpcld 22112 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c  A )  e.  CC )
7473, 62, 64divrecd 10106 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^c  A )  /  x
)  =  ( ( x  ^c  A )  x.  ( 1  /  x ) ) )
7570, 72, 743eqtrd 2477 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c 
( A  -  1 ) )  =  ( ( x  ^c  A )  x.  (
1  /  x ) ) )
7675oveq2d 6106 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^c  A )  x.  ( 1  /  x ) ) ) )
776rpcnd 11025 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
7865, 73, 77mul12d 9574 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^c  A )  x.  (
1  /  x ) ) )  =  ( ( x  ^c  A )  x.  ( A  x.  ( 1  /  x ) ) ) )
7973, 65, 77mulassd 9405 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( ( x  ^c  A )  x.  A )  x.  ( 1  /  x
) )  =  ( ( x  ^c  A )  x.  ( A  x.  ( 1  /  x ) ) ) )
8078, 79eqtr4d 2476 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8166oveq1d 6105 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8281oveq1d 6105 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( ( x  ^c  A )  x.  A )  x.  ( 1  /  x
) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8376, 80, 823eqtrd 2477 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8483mpteq2dva 4375 . 2  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( A  x.  ( x  ^c  ( A  - 
1 ) ) ) )  =  ( x  e.  RR+  |->  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) ) )
8560, 68, 843eqtr4d 2483 1  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  ^c  A ) ) )  =  ( x  e.  RR+  |->  ( A  x.  ( x  ^c 
( A  -  1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    i^i cin 3324    C_ wss 3325   {cpr 3876    e. cmpt 4347    |` cres 4838   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    - cmin 9591    / cdiv 9989   RR+crp 10987   expce 13343   TopOpenctopn 14356  ℂfldccnfld 17777  TopOnctopon 18458    _D cdv 21297   logclog 21965    ^c ccxp 21966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968
This theorem is referenced by:  dvsqr  22141
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