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Theorem rimul 9983
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
StepHypRef Expression
1 inelr 9982 . 2  |-  -.  _i  e.  RR
2 ax-icn 9041 . . . . . . 7  |-  _i  e.  CC
32a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  _i  e.  CC )
4 simpll 731 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  A  e.  RR )
54recnd 9106 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  A  e.  CC )
6 simpr 448 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  A  =/=  0 )
73, 5, 6divcan4d 9788 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( (
_i  x.  A )  /  A )  =  _i )
8 simplr 732 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( _i  x.  A )  e.  RR )
98, 4, 6redivcld 9834 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( (
_i  x.  A )  /  A )  e.  RR )
107, 9eqeltrrd 2510 . . . 4  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  _i  e.  RR )
1110ex 424 . . 3  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  ( A  =/=  0  ->  _i  e.  RR ) )
1211necon1bd 2666 . 2  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  ( -.  _i  e.  RR  ->  A  = 
0 ) )
131, 12mpi 17 1  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  A  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   _ici 8984    x. cmul 8987    / cdiv 9669
This theorem is referenced by:  cru  9984  cju  9988  crre  11911  tanarg  20506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670
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