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Theorem rimul 8310
Description: A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
rimul  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  A  =  0 )

Proof of Theorem rimul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inelr 8309 . . 3  |-  -.  _i  e.  RR
2 recexre 8303 . . . . . 6  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
32adantlr 466 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
4 simplll 505 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  A  e.  RR )
54recnd 7758 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  A  e.  CC )
6 simprl 503 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  x  e.  RR )
76recnd 7758 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  x  e.  CC )
8 ax-icn 7679 . . . . . . . . 9  |-  _i  e.  CC
9 mulass 7715 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( _i  x.  A
)  x.  x )  =  ( _i  x.  ( A  x.  x
) ) )
108, 9mp3an1 1285 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( _i  x.  A )  x.  x
)  =  ( _i  x.  ( A  x.  x ) ) )
115, 7, 10syl2anc 406 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  -> 
( ( _i  x.  A )  x.  x
)  =  ( _i  x.  ( A  x.  x ) ) )
12 oveq2 5748 . . . . . . . . 9  |-  ( ( A  x.  x )  =  1  ->  (
_i  x.  ( A  x.  x ) )  =  ( _i  x.  1 ) )
138mulid1i 7732 . . . . . . . . 9  |-  ( _i  x.  1 )  =  _i
1412, 13syl6eq 2164 . . . . . . . 8  |-  ( ( A  x.  x )  =  1  ->  (
_i  x.  ( A  x.  x ) )  =  _i )
1514ad2antll 480 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  -> 
( _i  x.  ( A  x.  x )
)  =  _i )
1611, 15eqtrd 2148 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  -> 
( ( _i  x.  A )  x.  x
)  =  _i )
17 simpllr 506 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  -> 
( _i  x.  A
)  e.  RR )
1817, 6remulcld 7760 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  -> 
( ( _i  x.  A )  x.  x
)  e.  RR )
1916, 18eqeltrrd 2193 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  _i  e.  RR )
203, 19rexlimddv 2529 . . . 4  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A #  0 )  ->  _i  e.  RR )
2120ex 114 . . 3  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  ( A #  0  ->  _i  e.  RR ) )
221, 21mtoi 636 . 2  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  -.  A #  0 )
23 0re 7730 . . . 4  |-  0  e.  RR
24 reapti 8304 . . . 4  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  =  0  <->  -.  A #  0 ) )
2523, 24mpan2 419 . . 3  |-  ( A  e.  RR  ->  ( A  =  0  <->  -.  A #  0
) )
2625adantr 272 . 2  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  ( A  =  0  <->  -.  A #  0 ) )
2722, 26mpbird 166 1  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  A  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392   class class class wbr 3897  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585   _ici 7586    x. cmul 7589   # creap 8299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-ltxr 7769  df-sub 7899  df-neg 7900  df-reap 8300
This theorem is referenced by:  rereim  8311  cru  8327  cju  8679  crre  10580
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