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Theorem 0cxpd 20558
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 20514 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 0  ^ c  A )  =  0 )
41, 2, 3syl2anc 643 1  |-  ( ph  ->  ( 0  ^ c  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2571  (class class class)co 6044   CCcc 8948   0cc0 8950    ^ c ccxp 20410
This theorem is referenced by:  cxpcn3lem  20588  cxpcn3  20589  cxpaddle  20593  cxpeq  20598  amgm  20786  abvcxp  21266  padicabvcxp  21283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-mulcl 9012  ax-i2m1 9018
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-cxp 20412
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