Theorem 0cxp 24412

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

Assertion

Ref	Expression
0cxp	|- ( ( A e. CC /\ A =/= 0 ) -> ( 0 ^c A ) = 0 )

Proof of Theorem 0cxp

Step	Hyp	Ref	Expression
1		0cn 10032	. . . 4 |- 0 e. CC
2		cxpval 24410	. . . 4 |- ( ( 0 e. CC /\ A e. CC ) -> ( 0 ^c A ) = if ( 0 = 0 , if ( A = 0 , 1 , 0 ) , ( exp ` ( A x. ( log ` 0 ) ) ) ) )
3	1, 2	mpan 706	. . 3 |- ( A e. CC -> ( 0 ^c A ) = if ( 0 = 0 , if ( A = 0 , 1 , 0 ) , ( exp ` ( A x. ( log ` 0 ) ) ) ) )
4		eqid 2622	. . . 4 |- 0 = 0
5	4	iftruei 4093	. . 3 |- if ( 0 = 0 , if ( A = 0 , 1 , 0 ) , ( exp ` ( A x. ( log ` 0 ) ) ) ) = if ( A = 0 , 1 , 0 )
6	3, 5	syl6eq 2672	. 2 |- ( A e. CC -> ( 0 ^c A ) = if ( A = 0 , 1 , 0 ) )
7		ifnefalse 4098	. 2 |- ( A =/= 0 -> if ( A = 0 , 1 , 0 ) = 0 )
8	6, 7	sylan9eq 2676	1 |- ( ( A e. CC /\ A =/= 0 ) -> ( 0 ^c A ) = 0 )

Syntax hints:   -> wi 4   /\ wa 384   = 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:  cxpexp  24414  cxpeq0  24424  cxpge0  24429  mulcxplem  24430  cxpmul2  24435  cxple2  24443  cxpsqrt  24449  0cxpd  24456  abscxpbnd  24494