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Theorem dvcxp1 20084
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 8830 . . . . 5  |-  RR  e.  _V
21prid1 3736 . . . 4  |-  RR  e.  { RR ,  CC }
32a1i 10 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
4 relogcl 19934 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
54adantl 452 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
6 rpreccl 10379 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
76adantl 452 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
8 recn 8829 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
9 mulcl 8823 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
10 efcl 12366 . . . . 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 5885 . . . 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 19986 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
16 relogf1o 19926 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
17 f1of 5474 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1816, 17mp1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1918feqmptd 5577 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
20 fvres 5544 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2120mpteq2ia 4104 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2219, 21syl6eq 2333 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2322oveq2d 5876 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2415, 23syl5reqr 2332 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
25 eqid 2285 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2625cnfldtopon 18294 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
27 toponmax 16668 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2826, 27mp1i 11 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
29 ax-resscn 8796 . . . . . 6  |-  RR  C_  CC
3029a1i 10 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
31 df-ss 3168 . . . . 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 8820 . . . . . . 7  |-  CC  e.  _V
3534prid2 3737 . . . . . 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 12366 . . . . . 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 8797 . . . . . . . 8  |-  1  e.  CC
4241a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4336dvmptid 19308 . . . . . . 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 19315 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
46 mulid1 8837 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4746mpteq2dv 4109 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4845, 47eqtrd 2317 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
49 eff 12365 . . . . . . . . . . 11  |-  exp : CC
--> CC
5049a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
5150feqmptd 5577 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
5251eqcomd 2290 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5352oveq2d 5876 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
54 dvef 19329 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5553, 54syl6eq 2333 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5655, 51eqtrd 2317 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
57 fveq2 5527 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5836, 36, 9, 37, 39, 39, 48, 56, 57, 57dvmptco 19323 . . . 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 19307 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
60 oveq2 5868 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
6160fveq2d 5531 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6261oveq1d 5875 . . 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 19323 . 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 10364 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6564adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
66 rpne0 10371 . . . . . 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 20061 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
7069mpteq2dva 4108 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^ c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
7170oveq2d 5876 . 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 20067 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  /  (
x  ^ c  1 ) ) )
7465cxp1d 20055 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
1 )  =  x )
7574oveq2d 5876 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  (
x  ^ c  1 ) )  =  ( ( x  ^ c  A )  /  x
) )
7665, 68cxpcld 20057 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  e.  CC )
7776, 65, 67divrecd 9541 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  x
)  =  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) )
7873, 75, 773eqtrd 2321 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  x.  (
1  /  x ) ) )
7978oveq2d 5876 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) ) )
807rpcnd 10394 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
8168, 76, 80mul12d 9023 . . . . 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 8860 . . . . 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 2320 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^ c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^ c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8469oveq1d 5875 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8584oveq1d 5875 . . . 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 2321 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8786mpteq2dva 4108 . 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 2327 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 1625    e. wcel 1686    =/= wne 2448   _Vcvv 2790    i^i cin 3153    C_ wss 3154   {cpr 3643    e. cmpt 4079    |` cres 4693   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    - cmin 9039    / cdiv 9425   RR+crp 10356   expce 12345   TopOpenctopn 13328  ℂfldccnfld 16379  TopOnctopon 16634    _D cdv 19215   logclog 19914    ^ c ccxp 19915
This theorem is referenced by:  dvsqr  20086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-cmp 17116  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917
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