MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvcxp1 Unicode version

Theorem dvcxp1 20078
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 8824 . . . . 5  |-  RR  e.  _V
21prid1 3735 . . . 4  |-  RR  e.  { RR ,  CC }
32a1i 10 . . 3  |-  ( A  e.  CC  ->  RR  e.  { RR ,  CC } )
4 relogcl 19928 . . . 4  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
54adantl 452 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( log `  x
)  e.  RR )
6 rpreccl 10373 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
76adantl 452 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  RR+ )
8 recn 8823 . . . 4  |-  ( y  e.  RR  ->  y  e.  CC )
9 mulcl 8817 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  ( A  x.  y
)  e.  CC )
10 efcl 12360 . . . . 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 5845 . . . 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 19980 . . . 4  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
16 relogf1o 19920 . . . . . . . 8  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
17 f1of 5438 . . . . . . . 8  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
1816, 17mp1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  ( log  |`  RR+ ) : RR+ --> RR )
1918feqmptd 5537 . . . . . 6  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) ) )
20 fvres 5503 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log  |`  RR+ ) `  x )  =  ( log `  x ) )
2120mpteq2ia 4103 . . . . . 6  |-  ( x  e.  RR+  |->  ( ( log  |`  RR+ ) `  x ) )  =  ( x  e.  RR+  |->  ( log `  x ) )
2219, 21syl6eq 2332 . . . . 5  |-  ( A  e.  CC  ->  ( log  |`  RR+ )  =  ( x  e.  RR+  |->  ( log `  x ) ) )
2322oveq2d 5836 . . . 4  |-  ( A  e.  CC  ->  ( RR  _D  ( log  |`  RR+ )
)  =  ( RR 
_D  ( x  e.  RR+  |->  ( log `  x
) ) ) )
2415, 23syl5reqr 2331 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( x  e.  RR+  |->  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) ) )
25 eqid 2284 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2625cnfldtopon 18288 . . . . 5  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
27 toponmax 16662 . . . . 5  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  e.  ( TopOpen ` fld ) )
2826, 27mp1i 11 . . . 4  |-  ( A  e.  CC  ->  CC  e.  ( TopOpen ` fld ) )
29 ax-resscn 8790 . . . . . 6  |-  RR  C_  CC
3029a1i 10 . . . . 5  |-  ( A  e.  CC  ->  RR  C_  CC )
31 df-ss 3167 . . . . 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 8814 . . . . . . 7  |-  CC  e.  _V
3534prid2 3736 . . . . . 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 12360 . . . . . 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 8791 . . . . . . . 8  |-  1  e.  CC
4241a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  CC )  ->  1  e.  CC )
4336dvmptid 19302 . . . . . . 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 19309 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  ( A  x.  1 ) ) )
46 mulid1 8831 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
4746mpteq2dv 4108 . . . . . 6  |-  ( A  e.  CC  ->  (
y  e.  CC  |->  ( A  x.  1 ) )  =  ( y  e.  CC  |->  A ) )
4845, 47eqtrd 2316 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( A  x.  y ) ) )  =  ( y  e.  CC  |->  A ) )
49 eff 12359 . . . . . . . . . . 11  |-  exp : CC
--> CC
5049a1i 10 . . . . . . . . . 10  |-  ( A  e.  CC  ->  exp : CC --> CC )
5150feqmptd 5537 . . . . . . . . 9  |-  ( A  e.  CC  ->  exp  =  ( x  e.  CC  |->  ( exp `  x
) ) )
5251eqcomd 2289 . . . . . . . 8  |-  ( A  e.  CC  ->  (
x  e.  CC  |->  ( exp `  x ) )  =  exp )
5352oveq2d 5836 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( CC  _D  exp ) )
54 dvef 19323 . . . . . . 7  |-  ( CC 
_D  exp )  =  exp
5553, 54syl6eq 2332 . . . . . 6  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  exp )
5655, 51eqtrd 2316 . . . . 5  |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( exp `  x
) ) )  =  ( x  e.  CC  |->  ( exp `  x ) ) )
57 fveq2 5486 . . . . 5  |-  ( x  =  ( A  x.  y )  ->  ( exp `  x )  =  ( exp `  ( A  x.  y )
) )
5836, 36, 9, 37, 39, 39, 48, 56, 57, 57dvmptco 19317 . . . 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 19301 . . 3  |-  ( A  e.  CC  ->  ( RR  _D  ( y  e.  RR  |->  ( exp `  ( A  x.  y )
) ) )  =  ( y  e.  RR  |->  ( ( exp `  ( A  x.  y )
)  x.  A ) ) )
60 oveq2 5828 . . . 4  |-  ( y  =  ( log `  x
)  ->  ( A  x.  y )  =  ( A  x.  ( log `  x ) ) )
6160fveq2d 5490 . . 3  |-  ( y  =  ( log `  x
)  ->  ( exp `  ( A  x.  y
) )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
6261oveq1d 5835 . . 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 19317 . 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 10358 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
6564adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  CC )
66 rpne0 10365 . . . . . 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 20055 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  =  ( exp `  ( A  x.  ( log `  x
) ) ) )
7069mpteq2dva 4107 . . 3  |-  ( A  e.  CC  ->  (
x  e.  RR+  |->  ( x  ^ c  A ) )  =  ( x  e.  RR+  |->  ( exp `  ( A  x.  ( log `  x ) ) ) ) )
7170oveq2d 5836 . 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 20061 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  /  (
x  ^ c  1 ) ) )
7465cxp1d 20049 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
1 )  =  x )
7574oveq2d 5836 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  (
x  ^ c  1 ) )  =  ( ( x  ^ c  A )  /  x
) )
7665, 68cxpcld 20051 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c  A )  e.  CC )
7776, 65, 67divrecd 9535 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  /  x
)  =  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) )
7873, 75, 773eqtrd 2320 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( x  ^ c 
( A  -  1 ) )  =  ( ( x  ^ c  A )  x.  (
1  /  x ) ) )
7978oveq2d 5836 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( A  x.  ( ( x  ^ c  A
)  x.  ( 1  /  x ) ) ) )
807rpcnd 10388 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( 1  /  x
)  e.  CC )
8168, 76, 80mul12d 9017 . . . . 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 8854 . . . . 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 2319 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
( x  ^ c  A )  x.  (
1  /  x ) ) )  =  ( ( ( x  ^ c  A )  x.  A
)  x.  ( 1  /  x ) ) )
8469oveq1d 5835 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( x  ^ c  A )  x.  A
)  =  ( ( exp `  ( A  x.  ( log `  x
) ) )  x.  A ) )
8584oveq1d 5835 . . . 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 2320 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( A  x.  (
x  ^ c  ( A  -  1 ) ) )  =  ( ( ( exp `  ( A  x.  ( log `  x ) ) )  x.  A )  x.  ( 1  /  x
) ) )
8786mpteq2dva 4107 . 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 2326 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 1685    =/= wne 2447   _Vcvv 2789    i^i cin 3152    C_ wss 3153   {cpr 3642    e. cmpt 4078    |` cres 4690   -->wf 5217   -1-1-onto->wf1o 5220   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    x. cmul 8738    - cmin 9033    / cdiv 9419   RR+crp 10350   expce 12339   TopOpenctopn 13322  ℂfldccnfld 16373  TopOnctopon 16628    _D cdv 19209   logclog 19908    ^ c ccxp 19909
This theorem is referenced by:  dvsqr  20080
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-cmp 17110  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cxp 19911
  Copyright terms: Public domain W3C validator