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Theorem dvcxp1 20579
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 9037 . . . . 5  |-  RR  e.  _V
21prid1 3872 . . . 4  |-  RR  e.  { RR ,  CC }
32a1i 11 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
4 relogcl 20426 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
54adantl 453 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
6 rpreccl 10591 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
76adantl 453 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
8 recn 9036 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
9 mulcl 9030 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
10 efcl 12640 . . . . 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 6065 . . . 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 20481 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
16 relogf1o 20417 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
17 f1of 5633 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1816, 17mp1i 12 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1918feqmptd 5738 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
20 fvres 5704 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2120mpteq2ia 4251 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2219, 21syl6eq 2452 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2322oveq2d 6056 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2415, 23syl5reqr 2451 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
25 eqid 2404 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2625cnfldtopon 18770 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
27 toponmax 16948 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2826, 27mp1i 12 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
29 ax-resscn 9003 . . . . . 6  |-  RR  C_  CC
3029a1i 11 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
31 df-ss 3294 . . . . 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 9027 . . . . . . 7  |-  CC  e.  _V
3534prid2 3873 . . . . . 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 12640 . . . . . 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 9004 . . . . . . . 8  |-  1  e.  CC
4241a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4336dvmptid 19796 . . . . . . 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 19803 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
46 mulid1 9044 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4746mpteq2dv 4256 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4845, 47eqtrd 2436 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
49 eff 12639 . . . . . . . . . . 11  |-  exp : CC
--> CC
5049a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
5150feqmptd 5738 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
5251eqcomd 2409 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5352oveq2d 6056 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
54 dvef 19817 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5553, 54syl6eq 2452 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5655, 51eqtrd 2436 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
57 fveq2 5687 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5836, 36, 9, 37, 39, 39, 48, 56, 57, 57dvmptco 19811 . . . 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 19795 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
60 oveq2 6048 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
6160fveq2d 5691 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6261oveq1d 6055 . . 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 19811 . 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 10576 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6564adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
66 rpne0 10583 . . . . . 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 20556 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
7069mpteq2dva 4255 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^ c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
7170oveq2d 6056 . 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 20562 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  /  (
x  ^ c  1 ) ) )
7465cxp1d 20550 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
1 )  =  x )
7574oveq2d 6056 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  (
x  ^ c  1 ) )  =  ( ( x  ^ c  A )  /  x
) )
7665, 68cxpcld 20552 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  e.  CC )
7776, 65, 67divrecd 9749 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  x
)  =  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) )
7873, 75, 773eqtrd 2440 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  x.  (
1  /  x ) ) )
7978oveq2d 6056 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) ) )
807rpcnd 10606 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
8168, 76, 80mul12d 9231 . . . . 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 9067 . . . . 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 2439 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^ c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^ c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8469oveq1d 6055 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8584oveq1d 6055 . . . 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 2440 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8786mpteq2dva 4255 . 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 2446 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 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    i^i cin 3279    C_ wss 3280   {cpr 3775    e. cmpt 4226    |` cres 4839   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    - cmin 9247    / cdiv 9633   RR+crp 10568   expce 12619   TopOpenctopn 13604  ℂfldccnfld 16658  TopOnctopon 16914    _D cdv 19703   logclog 20405    ^ c ccxp 20406
This theorem is referenced by:  dvsqr  20581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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