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Theorem rimul 8477
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 8476 . . 3  |-  -.  _i  e.  RR
2 recexre 8470 . . . . . 6  |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
32adantlr 469 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
4 simplll 523 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  A  e.  RR )
54recnd 7921 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  A  e.  CC )
6 simprl 521 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  x  e.  RR )
76recnd 7921 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  x  e.  CC )
8 ax-icn 7842 . . . . . . . . 9  |-  _i  e.  CC
9 mulass 7878 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC  /\  x  e.  CC )  ->  (
( _i  x.  A
)  x.  x )  =  ( _i  x.  ( A  x.  x
) ) )
108, 9mp3an1 1313 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( _i  x.  A )  x.  x
)  =  ( _i  x.  ( A  x.  x ) ) )
115, 7, 10syl2anc 409 . . . . . . 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 5847 . . . . . . . . 9  |-  ( ( A  x.  x )  =  1  ->  (
_i  x.  ( A  x.  x ) )  =  ( _i  x.  1 ) )
138mulid1i 7895 . . . . . . . . 9  |-  ( _i  x.  1 )  =  _i
1412, 13eqtrdi 2213 . . . . . . . 8  |-  ( ( A  x.  x )  =  1  ->  (
_i  x.  ( A  x.  x ) )  =  _i )
1514ad2antll 483 . . . . . . 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 2197 . . . . . 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 524 . . . . . . 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 7923 . . . . . 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 2242 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  /\  A #  0
)  /\  ( x  e.  RR  /\  ( A  x.  x )  =  1 ) )  ->  _i  e.  RR )
203, 19rexlimddv 2586 . . . 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 654 . 2  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  -.  A #  0 )
23 0re 7893 . . . 4  |-  0  e.  RR
24 reapti 8471 . . . 4  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  =  0  <->  -.  A #  0 ) )
2523, 24mpan2 422 . . 3  |-  ( A  e.  RR  ->  ( A  =  0  <->  -.  A #  0
) )
2625adantr 274 . 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 1342    e. wcel 2135   E.wrex 2443   class class class wbr 3979  (class class class)co 5839   CCcc 7745   RRcr 7746   0cc0 7747   1c1 7748   _ici 7749    x. cmul 7752   # creap 8466
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-cnex 7838  ax-resscn 7839  ax-1cn 7840  ax-1re 7841  ax-icn 7842  ax-addcl 7843  ax-addrcl 7844  ax-mulcl 7845  ax-mulrcl 7846  ax-addcom 7847  ax-mulcom 7848  ax-addass 7849  ax-mulass 7850  ax-distr 7851  ax-i2m1 7852  ax-0lt1 7853  ax-1rid 7854  ax-0id 7855  ax-rnegex 7856  ax-precex 7857  ax-cnre 7858  ax-pre-ltirr 7859  ax-pre-lttrn 7861  ax-pre-apti 7862  ax-pre-ltadd 7863  ax-pre-mulgt0 7864
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-iota 5150  df-fun 5187  df-fv 5193  df-riota 5795  df-ov 5842  df-oprab 5843  df-mpo 5844  df-pnf 7929  df-mnf 7930  df-ltxr 7932  df-sub 8065  df-neg 8066  df-reap 8467
This theorem is referenced by:  rereim  8478  cru  8494  cju  8850  crre  10793
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