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Theorem crne0 9925
Description: The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
crne0  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0 ) )

Proof of Theorem crne0
StepHypRef Expression
1 ax-icn 8982 . . . . . . . 8  |-  _i  e.  CC
21mul01i 9188 . . . . . . 7  |-  ( _i  x.  0 )  =  0
32oveq2i 6031 . . . . . 6  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
4 00id 9173 . . . . . 6  |-  ( 0  +  0 )  =  0
53, 4eqtri 2407 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
65eqeq2i 2397 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  =  ( 0  +  ( _i  x.  0 ) )  <->  ( A  +  ( _i  x.  B ) )  =  0 )
7 0re 9024 . . . . 5  |-  0  e.  RR
8 cru 9924 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  e.  RR  /\  0  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) )  =  ( 0  +  ( _i  x.  0 ) )  <-> 
( A  =  0  /\  B  =  0 ) ) )
97, 7, 8mpanr12 667 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  =  ( 0  +  ( _i  x.  0 ) )  <-> 
( A  =  0  /\  B  =  0 ) ) )
106, 9syl5bbr 251 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  =  0  <-> 
( A  =  0  /\  B  =  0 ) ) )
1110necon3abid 2583 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) )  =/=  0  <->  -.  ( A  =  0  /\  B  =  0 ) ) )
12 neorian 2637 . 2  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
1311, 12syl6rbbr 256 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  =/=  0  \/  B  =/=  0 )  <->  ( A  +  ( _i  x.  B ) )  =/=  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550  (class class class)co 6020   RRcr 8922   0cc0 8923   _ici 8925    + caddc 8926    x. cmul 8928
This theorem is referenced by:  crreczi  11431
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610
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