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Theorem dvcxp1 22195
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 9389 . . . 4  |-  RR  e.  { RR ,  CC }
21a1i 11 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
3 relogcl 22042 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
43adantl 466 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
5 rpreccl 11029 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
65adantl 466 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
7 recn 9387 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 9381 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
9 efcl 13383 . . . . 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 474 . . 3  |-  ( ( A  e.  CC  /\  y  e.  RR )  ->  ( exp `  ( A  x.  y )
)  e.  CC )
12 ovex 6131 . . . 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 22097 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
15 relogf1o 22033 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
16 f1of 5656 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1715, 16mp1i 12 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1817feqmptd 5759 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
19 fvres 5719 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2019mpteq2ia 4389 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2118, 20syl6eq 2491 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2221oveq2d 6122 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2314, 22syl5reqr 2490 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
24 eqid 2443 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2524cnfldtopon 20377 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
26 toponmax 18548 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2725, 26mp1i 12 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
28 ax-resscn 9354 . . . . . 6  |-  RR  C_  CC
2928a1i 11 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
30 df-ss 3357 . . . . 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 9390 . . . . . 6  |-  CC  e.  { RR ,  CC }
3433a1i 11 . . . . 5  |-  ( A  e.  CC  ->  CC  e.  { RR ,  CC } )
35 simpl 457 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  A  e.  CC )
36 efcl 13383 . . . . . 6  |-  ( x  e.  CC  ->  ( exp `  x )  e.  CC )
3736adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  x
)  e.  CC )
38 simpr 461 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  y  e.  CC )
39 1cnd 9417 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4034dvmptid 21446 . . . . . . 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 21453 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
43 mulid1 9398 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4443mpteq2dv 4394 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4542, 44eqtrd 2475 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
46 eff 13382 . . . . . . . . . . 11  |-  exp : CC
--> CC
4746a1i 11 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
4847feqmptd 5759 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
4948eqcomd 2448 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5049oveq2d 6122 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
51 dvef 21467 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5250, 51syl6eq 2491 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5352, 48eqtrd 2475 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
54 fveq2 5706 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5534, 34, 8, 35, 37, 37, 45, 53, 54, 54dvmptco 21461 . . . 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 21445 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
57 oveq2 6114 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
5857fveq2d 5710 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
5958oveq1d 6121 . . 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 21461 . 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 11014 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6261adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
63 rpne0 11021 . . . . . 6  |-  ( x  e.  RR+  ->  x  =/=  0 )
6463adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  =/=  0 )
65 simpl 457 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
6662, 64, 65cxpefd 22172 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6766mpteq2dva 4393 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
6867oveq2d 6122 . 2  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( x  ^c  A ) ) )  =  ( RR  _D  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) ) )
69 1cnd 9417 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
1  e.  CC )
7062, 64, 65, 69cxpsubd 22178 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c 
( A  -  1 ) )  =  ( ( x  ^c  A )  /  (
x  ^c  1 ) ) )
7162cxp1d 22166 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c 
1 )  =  x )
7271oveq2d 6122 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^c  A )  /  (
x  ^c  1 ) )  =  ( ( x  ^c  A )  /  x
) )
7362, 65cxpcld 22168 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c  A )  e.  CC )
7473, 62, 64divrecd 10125 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^c  A )  /  x
)  =  ( ( x  ^c  A )  x.  ( 1  /  x ) ) )
7570, 72, 743eqtrd 2479 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^c 
( A  -  1 ) )  =  ( ( x  ^c  A )  x.  (
1  /  x ) ) )
7675oveq2d 6122 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^c  A )  x.  ( 1  /  x ) ) ) )
776rpcnd 11044 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
7865, 73, 77mul12d 9593 . . . . 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 9424 . . . . 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 2478 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8166oveq1d 6121 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8281oveq1d 6121 . . . 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 2479 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8483mpteq2dva 4393 . 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 2485 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 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2987    i^i cin 3342    C_ wss 3343   {cpr 3894    e. cmpt 4365    |` cres 4857   -->wf 5429   -1-1-onto->wf1o 5432   ` cfv 5433  (class class class)co 6106   CCcc 9295   RRcr 9296   0cc0 9297   1c1 9298    x. cmul 9302    - cmin 9610    / cdiv 10008   RR+crp 11006   expce 13362   TopOpenctopn 14375  ℂfldccnfld 17833  TopOnctopon 18514    _D cdv 21353   logclog 22021    ^c ccxp 22022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ioc 11320  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-mod 11724  df-seq 11822  df-exp 11881  df-fac 12067  df-bc 12094  df-hash 12119  df-shft 12571  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-limsup 12964  df-clim 12981  df-rlim 12982  df-sum 13179  df-ef 13368  df-sin 13370  df-cos 13371  df-pi 13373  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-hom 14277  df-cco 14278  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-pt 14398  df-prds 14401  df-xrs 14455  df-qtop 14460  df-imas 14461  df-xps 14463  df-mre 14539  df-mrc 14540  df-acs 14542  df-mnd 15430  df-submnd 15480  df-mulg 15563  df-cntz 15850  df-cmn 16294  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-fbas 17829  df-fg 17830  df-cnfld 17834  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-cld 18638  df-ntr 18639  df-cls 18640  df-nei 18717  df-lp 18755  df-perf 18756  df-cn 18846  df-cnp 18847  df-haus 18934  df-cmp 19005  df-tx 19150  df-hmeo 19343  df-fil 19434  df-fm 19526  df-flim 19527  df-flf 19528  df-xms 19910  df-ms 19911  df-tms 19912  df-cncf 20469  df-limc 21356  df-dv 21357  df-log 22023  df-cxp 22024
This theorem is referenced by:  dvsqr  22197
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