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Theorem dvcxp1 20082
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 reex 8828 . . . . 5  |-  RR  e.  _V
21prid1 3734 . . . 4  |-  RR  e.  { RR ,  CC }
32a1i 10 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
4 relogcl 19932 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
54adantl 452 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
6 rpreccl 10377 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
76adantl 452 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
8 recn 8827 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
9 mulcl 8821 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
10 efcl 12364 . . . . 5  |-  ( ( A  x.  y )  e.  CC  ->  ( exp `  ( A  x.  y ) )  e.  CC )
119, 10syl 15 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
128, 11sylan2 460 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
13 ovex 5883 . . . 4  |-  ( ( exp `  ( A  x.  y ) )  x.  A )  e. 
_V
1413a1i 10 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
15 dvrelog 19984 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
16 relogf1o 19924 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
17 f1of 5472 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1816, 17mp1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1918feqmptd 5575 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
20 fvres 5542 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2120mpteq2ia 4102 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2219, 21syl6eq 2331 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2322oveq2d 5874 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2415, 23syl5reqr 2330 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
25 eqid 2283 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2625cnfldtopon 18292 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
27 toponmax 16666 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2826, 27mp1i 11 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
29 ax-resscn 8794 . . . . . 6  |-  RR  C_  CC
3029a1i 10 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
31 df-ss 3166 . . . . 5  |-  ( RR  C_  CC  <->  ( RR  i^i  CC )  =  RR )
3230, 31sylib 188 . . . 4  |-  ( A  e.  CC  ->  ( RR  i^i  CC )  =  RR )
3313a1i 10 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
34 cnex 8818 . . . . . . 7  |-  CC  e.  _V
3534prid2 3735 . . . . . 6  |-  CC  e.  { RR ,  CC }
3635a1i 10 . . . . 5  |-  ( A  e.  CC  ->  CC  e.  { RR ,  CC } )
37 simpl 443 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  A  e.  CC )
38 efcl 12364 . . . . . 6  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3938adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  x
)  e.  CC )
40 simpr 447 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  y  e.  CC )
41 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
4241a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4336dvmptid 19306 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  y ) )  =  ( y  e.  CC  |->  1 ) )
44 id 19 . . . . . . 7  |-  ( A  e.  CC  ->  A  e.  CC )
4536, 40, 42, 43, 44dvmptcmul 19313 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
46 mulid1 8835 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4746mpteq2dv 4107 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4845, 47eqtrd 2315 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
49 eff 12363 . . . . . . . . . . 11  |-  exp : CC
--> CC
5049a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
5150feqmptd 5575 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
5251eqcomd 2288 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5352oveq2d 5874 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
54 dvef 19327 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5553, 54syl6eq 2331 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5655, 51eqtrd 2315 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
57 fveq2 5525 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5836, 36, 9, 37, 39, 39, 48, 56, 57, 57dvmptco 19321 . . . 4  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  CC  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
5925, 3, 28, 32, 11, 33, 58dvmptres3 19305 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
60 oveq2 5866 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
6160fveq2d 5529 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6261oveq1d 5873 . . 3  |-  ( y  =  ( log `  x
)  ->  ( ( exp `  ( A  x.  y ) )  x.  A )  =  ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A ) )
633, 3, 5, 7, 12, 14, 24, 59, 61, 62dvmptco 19321 . 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
) ) ) )
64 rpcn 10362 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6564adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
66 rpne0 10369 . . . . . 6  |-  ( x  e.  RR+  ->  x  =/=  0 )
6766adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  =/=  0 )
68 simpl 443 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
6965, 67, 68cxpefd 20059 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
7069mpteq2dva 4106 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^ c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
7170oveq2d 5874 . 2  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  ^ c  A ) ) )  =  ( RR  _D  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) ) )
7241a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
1  e.  CC )
7365, 67, 68, 72cxpsubd 20065 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  /  (
x  ^ c  1 ) ) )
7465cxp1d 20053 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
1 )  =  x )
7574oveq2d 5874 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  (
x  ^ c  1 ) )  =  ( ( x  ^ c  A )  /  x
) )
7665, 68cxpcld 20055 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  e.  CC )
7776, 65, 67divrecd 9539 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  x
)  =  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) )
7873, 75, 773eqtrd 2319 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  x.  (
1  /  x ) ) )
7978oveq2d 5874 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) ) )
807rpcnd 10392 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
8168, 76, 80mul12d 9021 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^ c  A )  x.  (
1  /  x ) ) )  =  ( ( x  ^ c  A )  x.  ( A  x.  ( 1  /  x ) ) ) )
8276, 68, 80mulassd 8858 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( ( x  ^ c  A )  x.  A )  x.  ( 1  /  x
) )  =  ( ( x  ^ c  A )  x.  ( A  x.  ( 1  /  x ) ) ) )
8381, 82eqtr4d 2318 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^ c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^ c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8469oveq1d 5873 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8584oveq1d 5873 . . . 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
) ) )
8679, 83, 853eqtrd 2319 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8786mpteq2dva 4106 . 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
) ) ) )
8863, 71, 873eqtr4d 2325 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {cpr 3641    e. cmpt 4077    |` cres 4691   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037    / cdiv 9423   RR+crp 10354   expce 12343   TopOpenctopn 13326  ℂfldccnfld 16377  TopOnctopon 16632    _D cdv 19213   logclog 19912    ^ c ccxp 19913
This theorem is referenced by:  dvsqr  20084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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