Theorem cxpef 24411

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Assertion

Ref	Expression
cxpef	|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )

Proof of Theorem cxpef

Step	Hyp	Ref	Expression
1		cxpval 24410	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A ^c B ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )
2	1	3adant2 1080	. 2 |- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )
3		simp2 1062	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> A =/= 0 )
4	3	neneqd 2799	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> -. A = 0 )
5	4	iffalsed 4097	. 2 |- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( B x. ( log ` A ) ) ) )
6	2, 5	eqtrd 2656	1 |- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   expce 14792   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cxp 24304

This theorem is referenced by:  cxpexpz  24413  logcxp  24415  1cxp  24418  ecxp  24419  rpcxpcl  24422  cxpne0  24423  cxpadd  24425  mulcxp  24431  cxpmul  24434  abscxp  24438  abscxp2  24439  cxplt  24440  cxple2  24443  cxpsqrtlem  24448  cxpsqrt  24449  cxpefd  24458  1cubrlem  24568  bposlem9  25017  iexpire  31621