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Theorem 0cxpd 20632
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 20588 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )
41, 2, 3syl2anc 644 1  |-  ( ph  ->  ( 0  ^ c  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727    =/= wne 2605  (class class class)co 6110   CCcc 9019   0cc0 9021    ^ c ccxp 20484
This theorem is referenced by:  cxpcn3lem  20662  cxpcn3  20663  cxpaddle  20667  cxpeq  20672  amgm  20860  abvcxp  21340  padicabvcxp  21357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-mulcl 9083  ax-i2m1 9089
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-cxp 20486
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