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Theorem rimul 9925
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 9924 . 2  |-  -.  _i  e.  RR
2 ax-icn 8984 . . . . . . 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 9049 . . . . . 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 9730 . . . . 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 9776 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( (
_i  x.  A )  /  A )  e.  RR )
107, 9eqeltrrd 2464 . . . 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 2620 . 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 1649    e. wcel 1717    =/= wne 2552  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   _ici 8927    x. cmul 8930    / cdiv 9611
This theorem is referenced by:  cru  9926  cju  9930  crre  11848  tanarg  20383
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612
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