Theorem 0cxp 22116

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 9383	. . . 4 |- 0 e. CC
2		cxpval 22114	. . . 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 670	. . 3 |- ( A e. CC -> ( 0 ^c A ) = if ( 0 = 0 , if ( A = 0 , 1 , 0 ) , ( exp ` ( A x. ( log ` 0 ) ) ) ) )
4		eqid 2443	. . . 4 |- 0 = 0
5	4	iftruei 3803	. . 3 |- if ( 0 = 0 , if ( A = 0 , 1 , 0 ) , ( exp ` ( A x. ( log ` 0 ) ) ) ) = if ( A = 0 , 1 , 0 )
6	3, 5	syl6eq 2491	. 2 |- ( A e. CC -> ( 0 ^c A ) = if ( A = 0 , 1 , 0 ) )
7		ifnefalse 3806	. 2 |- ( A =/= 0 -> if ( A = 0 , 1 , 0 ) = 0 )
8	6, 7	sylan9eq 2495	1 |- ( ( A e. CC /\ A =/= 0 ) -> ( 0 ^c A ) = 0 )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2611   ifcif 3796   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288   x. cmul 9292   expce 13352   logclog 22011   ^c ccxp 22012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-mulcl 9349  ax-i2m1 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-cxp 22014

This theorem is referenced by:  cxpexp  22118  cxpeq0  22128  cxpge0  22133  mulcxplem  22134  cxpmul2  22139  cxple2  22147  cxpsqr  22153  0cxpd  22160  abscxpbnd  22196