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

Theorem 0cxpd 22957
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 22913 . 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 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6295   CCcc 9502   0cc0 9504    ^c ccxp 22809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-mulcl 9566  ax-i2m1 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-cxp 22811
This theorem is referenced by:  cxpcn3lem  22987  cxpcn3  22988  cxpaddle  22992  cxpeq  22997  amgm  23186  abvcxp  23666  padicabvcxp  23683
  Copyright terms: Public domain W3C validator