MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0cxpd Unicode version

Theorem 0cxpd 20168
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 20124 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )
41, 2, 3syl2anc 642 1  |-  ( ph  ->  ( 0  ^ c  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    =/= wne 2521  (class class class)co 5945   CCcc 8825   0cc0 8827    ^ c ccxp 20020
This theorem is referenced by:  cxpcn3lem  20198  cxpcn3  20199  cxpaddle  20203  cxpeq  20208  amgm  20396  abvcxp  20876  padicabvcxp  20893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-mulcl 8889  ax-i2m1 8895
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-cxp 20022
  Copyright terms: Public domain W3C validator