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Theorem 0cxpd 22960
Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
Hypotheses
Ref Expression
cxp0d.1  |-  ( ph  ->  A  e.  CC )
cxpefd.2  |-  ( ph  ->  A  =/=  0 )
Assertion
Ref Expression
0cxpd  |-  ( ph  ->  ( 0  ^c  A )  =  0 )

Proof of Theorem 0cxpd
StepHypRef Expression
1 cxp0d.1 . 2  |-  ( ph  ->  A  e.  CC )
2 cxpefd.2 . 2  |-  ( ph  ->  A  =/=  0 )
3 0cxp 22916 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^c  A )  =  0 )
41, 2, 3syl2anc 661 1  |-  ( ph  ->  ( 0  ^c  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802    =/= wne 2636  (class class class)co 6278   CCcc 9490   0cc0 9492    ^c ccxp 22812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pr 4673  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-mulcl 9554  ax-i2m1 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-iota 5538  df-fun 5577  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-cxp 22814
This theorem is referenced by:  cxpcn3lem  22990  cxpcn3  22991  cxpaddle  22995  cxpeq  23000  amgm  23189  abvcxp  23669  padicabvcxp  23686
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