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Theorem rimul 10516
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 10515 . 2  |-  -.  _i  e.  RR
2 ax-icn 9540 . . . . . . 7  |-  _i  e.  CC
32a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  _i  e.  CC )
4 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  A  e.  RR )
54recnd 9611 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  A  e.  CC )
6 simpr 461 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  A  =/=  0 )
73, 5, 6divcan4d 10315 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( (
_i  x.  A )  /  A )  =  _i )
8 simplr 754 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( _i  x.  A )  e.  RR )
98, 4, 6redivcld 10361 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  ( (
_i  x.  A )  /  A )  e.  RR )
107, 9eqeltrrd 2549 . . . 4  |-  ( ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  /\  A  =/=  0
)  ->  _i  e.  RR )
1110ex 434 . . 3  |-  ( ( A  e.  RR  /\  ( _i  x.  A
)  e.  RR )  ->  ( A  =/=  0  ->  _i  e.  RR ) )
1211necon1bd 2678 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   _ici 9483    x. cmul 9486    / cdiv 10195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196
This theorem is referenced by:  cru  10517  cju  10521  crre  12897  tanarg  22725
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