Theorem dvcncxp1 24484

Description: Derivative of complex power with respect to first argument on the complex plane. (Contributed by Brendan Leahy, 18-Dec-2018.)

Hypothesis

Ref	Expression
dvcncxp1.d	|- D = ( CC \ ( -oo (,] 0 ) )

Assertion

Ref	Expression
dvcncxp1	|- ( A e. CC -> ( CC _D ( x e. D |-> ( x ^c A ) ) ) = ( x e. D |-> ( A x. ( x ^c ( A - 1 ) ) ) ) )

Distinct variable groups:   x, A   x, D

Proof of Theorem dvcncxp1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnelprrecn 10029	. . . 4 |- CC e. { RR , CC }
2	1	a1i 11	. . 3 |- ( A e. CC -> CC e. { RR , CC } )
3		dvcncxp1.d	. . . . . . 7 |- D = ( CC \ ( -oo (,] 0 ) )
4		difss 3737	. . . . . . 7 |- ( CC \ ( -oo (,] 0 ) ) C_ CC
5	3, 4	eqsstri 3635	. . . . . 6 |- D C_ CC
6	5	sseli 3599	. . . . 5 |- ( x e. D -> x e. CC )
7	3	logdmn0 24386	. . . . 5 |- ( x e. D -> x =/= 0 )
8	6, 7	logcld 24317	. . . 4 |- ( x e. D -> ( log ` x ) e. CC )
9	8	adantl 482	. . 3 |- ( ( A e. CC /\ x e. D ) -> ( log ` x ) e. CC )
10	6, 7	reccld 10794	. . . 4 |- ( x e. D -> ( 1 / x ) e. CC )
11	10	adantl 482	. . 3 |- ( ( A e. CC /\ x e. D ) -> ( 1 / x ) e. CC )
12		mulcl 10020	. . . 4 |- ( ( A e. CC /\ y e. CC ) -> ( A x. y ) e. CC )
13		efcl 14813	. . . 4 |- ( ( A x. y ) e. CC -> ( exp ` ( A x. y ) ) e. CC )
14	12, 13	syl 17	. . 3 |- ( ( A e. CC /\ y e. CC ) -> ( exp ` ( A x. y ) ) e. CC )
15		ovexd 6680	. . 3 |- ( ( A e. CC /\ y e. CC ) -> ( ( exp ` ( A x. y ) ) x. A ) e. _V )
16	3	dvlog 24397	. . . 4 |- ( CC _D ( log |` D ) ) = ( x e. D |-> ( 1 / x ) )
17	3	logcn 24393	. . . . . . . 8 |- ( log |` D ) e. ( D -cn-> CC )
18		cncff 22696	. . . . . . . 8 |- ( ( log |` D ) e. ( D -cn-> CC ) -> ( log |` D ) : D --> CC )
19	17, 18	mp1i 13	. . . . . . 7 |- ( A e. CC -> ( log |` D ) : D --> CC )
20	19	feqmptd 6249	. . . . . 6 |- ( A e. CC -> ( log |` D ) = ( x e. D |-> ( ( log |` D ) ` x ) ) )
21		fvres 6207	. . . . . . 7 |- ( x e. D -> ( ( log |` D ) ` x ) = ( log ` x ) )
22	21	mpteq2ia 4740	. . . . . 6 |- ( x e. D |-> ( ( log |` D ) ` x ) ) = ( x e. D |-> ( log ` x ) )
23	20, 22	syl6eq 2672	. . . . 5 |- ( A e. CC -> ( log |` D ) = ( x e. D |-> ( log ` x ) ) )
24	23	oveq2d 6666	. . . 4 |- ( A e. CC -> ( CC _D ( log |` D ) ) = ( CC _D ( x e. D |-> ( log ` x ) ) ) )
25	16, 24	syl5reqr 2671	. . 3 |- ( A e. CC -> ( CC _D ( x e. D |-> ( log ` x ) ) ) = ( x e. D |-> ( 1 / x ) ) )
26		simpl 473	. . . 4 |- ( ( A e. CC /\ y e. CC ) -> A e. CC )
27		efcl 14813	. . . . 5 |- ( x e. CC -> ( exp ` x ) e. CC )
28	27	adantl 482	. . . 4 |- ( ( A e. CC /\ x e. CC ) -> ( exp ` x ) e. CC )
29		simpr 477	. . . . . 6 |- ( ( A e. CC /\ y e. CC ) -> y e. CC )
30		1cnd 10056	. . . . . 6 |- ( ( A e. CC /\ y e. CC ) -> 1 e. CC )
31	2	dvmptid 23720	. . . . . 6 |- ( A e. CC -> ( CC _D ( y e. CC |-> y ) ) = ( y e. CC |-> 1 ) )
32		id 22	. . . . . 6 |- ( A e. CC -> A e. CC )
33	2, 29, 30, 31, 32	dvmptcmul 23727	. . . . 5 |- ( A e. CC -> ( CC _D ( y e. CC |-> ( A x. y ) ) ) = ( y e. CC |-> ( A x. 1 ) ) )
34		mulid1 10037	. . . . . 6 |- ( A e. CC -> ( A x. 1 ) = A )
35	34	mpteq2dv 4745	. . . . 5 |- ( A e. CC -> ( y e. CC |-> ( A x. 1 ) ) = ( y e. CC |-> A ) )
36	33, 35	eqtrd 2656	. . . 4 |- ( A e. CC -> ( CC _D ( y e. CC |-> ( A x. y ) ) ) = ( y e. CC |-> A ) )
37		dvef 23743	. . . . 5 |- ( CC _D exp ) = exp
38		eff 14812	. . . . . . . 8 |- exp : CC --> CC
39	38	a1i 11	. . . . . . 7 |- ( A e. CC -> exp : CC --> CC )
40	39	feqmptd 6249	. . . . . 6 |- ( A e. CC -> exp = ( x e. CC |-> ( exp ` x ) ) )
41	40	oveq2d 6666	. . . . 5 |- ( A e. CC -> ( CC _D exp ) = ( CC _D ( x e. CC |-> ( exp ` x ) ) ) )
42	37, 41, 40	3eqtr3a 2680	. . . 4 |- ( A e. CC -> ( CC _D ( x e. CC |-> ( exp ` x ) ) ) = ( x e. CC |-> ( exp ` x ) ) )
43		fveq2 6191	. . . 4 |- ( x = ( A x. y ) -> ( exp ` x ) = ( exp ` ( A x. y ) ) )
44	2, 2, 12, 26, 28, 28, 36, 42, 43, 43	dvmptco 23735	. . 3 |- ( A e. CC -> ( CC _D ( y e. CC |-> ( exp ` ( A x. y ) ) ) ) = ( y e. CC |-> ( ( exp ` ( A x. y ) ) x. A ) ) )
45		oveq2 6658	. . . 4 |- ( y = ( log ` x ) -> ( A x. y ) = ( A x. ( log ` x ) ) )
46	45	fveq2d 6195	. . 3 |- ( y = ( log ` x ) -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. ( log ` x ) ) ) )
47	46	oveq1d 6665	. . 3 |- ( y = ( log ` x ) -> ( ( exp ` ( A x. y ) ) x. A ) = ( ( exp ` ( A x. ( log ` x ) ) ) x. A ) )
48	2, 2, 9, 11, 14, 15, 25, 44, 46, 47	dvmptco 23735	. 2 |- ( A e. CC -> ( CC _D ( x e. D |-> ( exp ` ( A x. ( log ` x ) ) ) ) ) = ( x e. D |-> ( ( ( exp ` ( A x. ( log ` x ) ) ) x. A ) x. ( 1 / x ) ) ) )
49	6	adantl 482	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> x e. CC )
50	7	adantl 482	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> x =/= 0 )
51		simpl 473	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> A e. CC )
52	49, 50, 51	cxpefd 24458	. . . 4 |- ( ( A e. CC /\ x e. D ) -> ( x ^c A ) = ( exp ` ( A x. ( log ` x ) ) ) )
53	52	mpteq2dva 4744	. . 3 |- ( A e. CC -> ( x e. D |-> ( x ^c A ) ) = ( x e. D |-> ( exp ` ( A x. ( log ` x ) ) ) ) )
54	53	oveq2d 6666	. 2 |- ( A e. CC -> ( CC _D ( x e. D |-> ( x ^c A ) ) ) = ( CC _D ( x e. D |-> ( exp ` ( A x. ( log ` x ) ) ) ) ) )
55		1cnd 10056	. . . . . . 7 |- ( ( A e. CC /\ x e. D ) -> 1 e. CC )
56	49, 50, 51, 55	cxpsubd 24464	. . . . . 6 |- ( ( A e. CC /\ x e. D ) -> ( x ^c ( A - 1 ) ) = ( ( x ^c A ) / ( x ^c 1 ) ) )
57	49	cxp1d 24452	. . . . . . 7 |- ( ( A e. CC /\ x e. D ) -> ( x ^c 1 ) = x )
58	57	oveq2d 6666	. . . . . 6 |- ( ( A e. CC /\ x e. D ) -> ( ( x ^c A ) / ( x ^c 1 ) ) = ( ( x ^c A ) / x ) )
59	49, 51	cxpcld 24454	. . . . . . 7 |- ( ( A e. CC /\ x e. D ) -> ( x ^c A ) e. CC )
60	59, 49, 50	divrecd 10804	. . . . . 6 |- ( ( A e. CC /\ x e. D ) -> ( ( x ^c A ) / x ) = ( ( x ^c A ) x. ( 1 / x ) ) )
61	56, 58, 60	3eqtrd 2660	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> ( x ^c ( A - 1 ) ) = ( ( x ^c A ) x. ( 1 / x ) ) )
62	61	oveq2d 6666	. . . 4 |- ( ( A e. CC /\ x e. D ) -> ( A x. ( x ^c ( A - 1 ) ) ) = ( A x. ( ( x ^c A ) x. ( 1 / x ) ) ) )
63	51, 59, 11	mul12d 10245	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> ( A x. ( ( x ^c A ) x. ( 1 / x ) ) ) = ( ( x ^c A ) x. ( A x. ( 1 / x ) ) ) )
64	59, 51, 11	mulassd 10063	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> ( ( ( x ^c A ) x. A ) x. ( 1 / x ) ) = ( ( x ^c A ) x. ( A x. ( 1 / x ) ) ) )
65	63, 64	eqtr4d 2659	. . . 4 |- ( ( A e. CC /\ x e. D ) -> ( A x. ( ( x ^c A ) x. ( 1 / x ) ) ) = ( ( ( x ^c A ) x. A ) x. ( 1 / x ) ) )
66	52	oveq1d 6665	. . . . 5 |- ( ( A e. CC /\ x e. D ) -> ( ( x ^c A ) x. A ) = ( ( exp ` ( A x. ( log ` x ) ) ) x. A ) )
67	66	oveq1d 6665	. . . 4 |- ( ( A e. CC /\ x e. D ) -> ( ( ( x ^c A ) x. A ) x. ( 1 / x ) ) = ( ( ( exp ` ( A x. ( log ` x ) ) ) x. A ) x. ( 1 / x ) ) )
68	62, 65, 67	3eqtrd 2660	. . 3 |- ( ( A e. CC /\ x e. D ) -> ( A x. ( x ^c ( A - 1 ) ) ) = ( ( ( exp ` ( A x. ( log ` x ) ) ) x. A ) x. ( 1 / x ) ) )
69	68	mpteq2dva 4744	. 2 |- ( A e. CC -> ( x e. D |-> ( A x. ( x ^c ( A - 1 ) ) ) ) = ( x e. D |-> ( ( ( exp ` ( A x. ( log ` x ) ) ) x. A ) x. ( 1 / x ) ) ) )
70	48, 54, 69	3eqtr4d 2666	1 |- ( A e. CC -> ( CC _D ( x e. D |-> ( x ^c A ) ) ) = ( x e. D |-> ( A x. ( x ^c ( A - 1 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   {cpr 4179   |-> cmpt 4729   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   -oocmnf 10072   - cmin 10266   / cdiv 10684   (,]cioc 12176   expce 14792   -cn->ccncf 22679   _D cdv 23627   logclog 24301   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  dvcnsqrt  24485  binomcxplemdvbinom  38552