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Theorem dvcxp1 20614
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 9070 . . . . 5  |-  RR  e.  _V
21prid1 3904 . . . 4  |-  RR  e.  { RR ,  CC }
32a1i 11 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
4 relogcl 20461 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
54adantl 453 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
6 rpreccl 10624 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
76adantl 453 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
8 recn 9069 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
9 mulcl 9063 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
10 efcl 12673 . . . . 5  |-  ( ( A  x.  y )  e.  CC  ->  ( exp `  ( A  x.  y ) )  e.  CC )
119, 10syl 16 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
128, 11sylan2 461 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
13 ovex 6097 . . . 4  |-  ( ( exp `  ( A  x.  y ) )  x.  A )  e. 
_V
1413a1i 11 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
15 dvrelog 20516 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
16 relogf1o 20452 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
17 f1of 5665 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1816, 17mp1i 12 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1918feqmptd 5770 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
20 fvres 5736 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2120mpteq2ia 4283 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2219, 21syl6eq 2483 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2322oveq2d 6088 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2415, 23syl5reqr 2482 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
25 eqid 2435 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2625cnfldtopon 18805 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
27 toponmax 16981 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2826, 27mp1i 12 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
29 ax-resscn 9036 . . . . . 6  |-  RR  C_  CC
3029a1i 11 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
31 df-ss 3326 . . . . 5  |-  ( RR  C_  CC  <->  ( RR  i^i  CC )  =  RR )
3230, 31sylib 189 . . . 4  |-  ( A  e.  CC  ->  ( RR  i^i  CC )  =  RR )
3313a1i 11 . . . 4  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( ( exp `  ( A  x.  y )
)  x.  A )  e.  _V )
34 cnex 9060 . . . . . . 7  |-  CC  e.  _V
3534prid2 3905 . . . . . 6  |-  CC  e.  { RR ,  CC }
3635a1i 11 . . . . 5  |-  ( A  e.  CC  ->  CC  e.  { RR ,  CC } )
37 simpl 444 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  A  e.  CC )
38 efcl 12673 . . . . . 6  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3938adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  x
)  e.  CC )
40 simpr 448 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  y  e.  CC )
41 ax-1cn 9037 . . . . . . . 8  |-  1  e.  CC
4241a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4336dvmptid 19831 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  y ) )  =  ( y  e.  CC  |->  1 ) )
44 id 20 . . . . . . 7  |-  ( A  e.  CC  ->  A  e.  CC )
4536, 40, 42, 43, 44dvmptcmul 19838 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
46 mulid1 9077 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4746mpteq2dv 4288 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4845, 47eqtrd 2467 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
49 eff 12672 . . . . . . . . . . 11  |-  exp : CC
--> CC
5049a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
5150feqmptd 5770 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
5251eqcomd 2440 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5352oveq2d 6088 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
54 dvef 19852 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5553, 54syl6eq 2483 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5655, 51eqtrd 2467 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
57 fveq2 5719 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5836, 36, 9, 37, 39, 39, 48, 56, 57, 57dvmptco 19846 . . . 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 19830 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
60 oveq2 6080 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
6160fveq2d 5723 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6261oveq1d 6087 . . 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 19846 . 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 10609 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6564adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
66 rpne0 10616 . . . . . 6  |-  ( x  e.  RR+  ->  x  =/=  0 )
6766adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  =/=  0 )
68 simpl 444 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
6965, 67, 68cxpefd 20591 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
7069mpteq2dva 4287 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^ c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
7170oveq2d 6088 . 2  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  ^ c  A ) ) )  =  ( RR  _D  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) ) )
7241a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
1  e.  CC )
7365, 67, 68, 72cxpsubd 20597 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  /  (
x  ^ c  1 ) ) )
7465cxp1d 20585 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
1 )  =  x )
7574oveq2d 6088 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  (
x  ^ c  1 ) )  =  ( ( x  ^ c  A )  /  x
) )
7665, 68cxpcld 20587 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  e.  CC )
7776, 65, 67divrecd 9782 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  x
)  =  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) )
7873, 75, 773eqtrd 2471 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  x.  (
1  /  x ) ) )
7978oveq2d 6088 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) ) )
807rpcnd 10639 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
8168, 76, 80mul12d 9264 . . . . 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 9100 . . . . 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 2470 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^ c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^ c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8469oveq1d 6087 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8584oveq1d 6087 . . . 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 2471 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8786mpteq2dva 4287 . 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 2477 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 359    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {cpr 3807    e. cmpt 4258    |` cres 4871   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    x. cmul 8984    - cmin 9280    / cdiv 9666   RR+crp 10601   expce 12652   TopOpenctopn 13637  ℂfldccnfld 16691  TopOnctopon 16947    _D cdv 19738   logclog 20440    ^ c ccxp 20441
This theorem is referenced by:  dvsqr  20616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-sum 12468  df-ef 12658  df-sin 12660  df-cos 12661  df-pi 12663  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-cmp 17438  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442  df-cxp 20443
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