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Theorem crap0 8946
Description: The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
Assertion
Ref Expression
crap0  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A #  0  \/  B #  0 )  <-> 
( A  +  ( _i  x.  B ) ) #  0 ) )

Proof of Theorem crap0
StepHypRef Expression
1 0re 7988 . . 3  |-  0  e.  RR
2 apreim 8591 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  e.  RR  /\  0  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
31, 1, 2mpanr12 439 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
4 ax-icn 7937 . . . . . 6  |-  _i  e.  CC
54mul01i 8379 . . . . 5  |-  ( _i  x.  0 )  =  0
65oveq2i 5908 . . . 4  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
7 00id 8129 . . . 4  |-  ( 0  +  0 )  =  0
86, 7eqtri 2210 . . 3  |-  ( 0  +  ( _i  x.  0 ) )  =  0
98breq2i 4026 . 2  |-  ( ( A  +  ( _i  x.  B ) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A  +  ( _i  x.  B
) ) #  0 )
103, 9bitr3di 195 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A #  0  \/  B #  0 )  <-> 
( A  +  ( _i  x.  B ) ) #  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   RRcr 7841   0cc0 7842   _ici 7844    + caddc 7845    x. cmul 7847   # cap 8569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-ltxr 8028  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570
This theorem is referenced by:  abs00ap  11106
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