Theorem cxpval 24410

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

Assertion

Ref	Expression
cxpval	|- ( ( A e. CC /\ B e. CC ) -> ( A ^c B ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )

Proof of Theorem cxpval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( x = A /\ y = B ) -> x = A )
2	1	eqeq1d 2624	. . 3 |- ( ( x = A /\ y = B ) -> ( x = 0 <-> A = 0 ) )
3		simpr 477	. . . . 5 |- ( ( x = A /\ y = B ) -> y = B )
4	3	eqeq1d 2624	. . . 4 |- ( ( x = A /\ y = B ) -> ( y = 0 <-> B = 0 ) )
5	4	ifbid 4108	. . 3 |- ( ( x = A /\ y = B ) -> if ( y = 0 , 1 , 0 ) = if ( B = 0 , 1 , 0 ) )
6	1	fveq2d 6195	. . . . 5 |- ( ( x = A /\ y = B ) -> ( log ` x ) = ( log ` A ) )
7	3, 6	oveq12d 6668	. . . 4 |- ( ( x = A /\ y = B ) -> ( y x. ( log ` x ) ) = ( B x. ( log ` A ) ) )
8	7	fveq2d 6195	. . 3 |- ( ( x = A /\ y = B ) -> ( exp ` ( y x. ( log ` x ) ) ) = ( exp ` ( B x. ( log ` A ) ) ) )
9	2, 5, 8	ifbieq12d 4113	. 2 |- ( ( x = A /\ y = B ) -> if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )
10		df-cxp 24304	. 2 |- ^c = ( x e. CC , y e. CC |-> if ( x = 0 , if ( y = 0 , 1 , 0 ) , ( exp ` ( y x. ( log ` x ) ) ) ) )
11		ax-1cn 9994	. . . . 5 |- 1 e. CC
12		0cn 10032	. . . . 5 |- 0 e. CC
13	11, 12	keepel 4155	. . . 4 |- if ( B = 0 , 1 , 0 ) e. CC
14	13	elexi 3213	. . 3 |- if ( B = 0 , 1 , 0 ) e. _V
15		fvex 6201	. . 3 |- ( exp ` ( B x. ( log ` A ) ) ) e. _V
16	14, 15	ifex 4156	. 2 |- if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) e. _V
17	9, 10, 16	ovmpt2a 6791	1 |- ( ( A e. CC /\ B e. CC ) -> ( A ^c B ) = if ( A = 0 , if ( B = 0 , 1 , 0 ) , ( exp ` ( B x. ( log ` A ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   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-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:  cxpef  24411  0cxp  24412  cxpexp  24414  cxpcl  24420  recxpcl  24421