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Theorem rennim 11046
Description: A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)
Assertion
Ref Expression
rennim  |-  ( A  e.  RR  ->  (
_i  x.  A )  e/  RR+ )

Proof of Theorem rennim
StepHypRef Expression
1 ax-icn 7937 . . . . . . 7  |-  _i  e.  CC
2 recn 7975 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
3 mulcl 7969 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 414 . . . . . 6  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
5 rpre 9692 . . . . . . 7  |-  ( ( _i  x.  A )  e.  RR+  ->  ( _i  x.  A )  e.  RR )
6 rereb 10907 . . . . . . 7  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  e.  RR  <->  ( Re `  ( _i  x.  A
) )  =  ( _i  x.  A ) ) )
75, 6imbitrid 154 . . . . . 6  |-  ( ( _i  x.  A )  e.  CC  ->  (
( _i  x.  A
)  e.  RR+  ->  ( Re `  ( _i  x.  A ) )  =  ( _i  x.  A ) ) )
84, 7syl 14 . . . . 5  |-  ( A  e.  RR  ->  (
( _i  x.  A
)  e.  RR+  ->  ( Re `  ( _i  x.  A ) )  =  ( _i  x.  A ) ) )
94addlidd 8138 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  +  ( _i  x.  A ) )  =  ( _i  x.  A ) )
109fveq2d 5538 . . . . . . 7  |-  ( A  e.  RR  ->  (
Re `  ( 0  +  ( _i  x.  A ) ) )  =  ( Re `  ( _i  x.  A
) ) )
11 0re 7988 . . . . . . . 8  |-  0  e.  RR
12 crre 10901 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( Re `  (
0  +  ( _i  x.  A ) ) )  =  0 )
1311, 12mpan 424 . . . . . . 7  |-  ( A  e.  RR  ->  (
Re `  ( 0  +  ( _i  x.  A ) ) )  =  0 )
1410, 13eqtr3d 2224 . . . . . 6  |-  ( A  e.  RR  ->  (
Re `  ( _i  x.  A ) )  =  0 )
1514eqeq1d 2198 . . . . 5  |-  ( A  e.  RR  ->  (
( Re `  (
_i  x.  A )
)  =  ( _i  x.  A )  <->  0  =  ( _i  x.  A
) ) )
168, 15sylibd 149 . . . 4  |-  ( A  e.  RR  ->  (
( _i  x.  A
)  e.  RR+  ->  0  =  ( _i  x.  A ) ) )
17 rpne0 9701 . . . . . 6  |-  ( ( _i  x.  A )  e.  RR+  ->  ( _i  x.  A )  =/=  0 )
1817necon2bi 2415 . . . . 5  |-  ( ( _i  x.  A )  =  0  ->  -.  ( _i  x.  A
)  e.  RR+ )
1918eqcoms 2192 . . . 4  |-  ( 0  =  ( _i  x.  A )  ->  -.  ( _i  x.  A
)  e.  RR+ )
2016, 19syl6 33 . . 3  |-  ( A  e.  RR  ->  (
( _i  x.  A
)  e.  RR+  ->  -.  ( _i  x.  A
)  e.  RR+ )
)
2120pm2.01d 619 . 2  |-  ( A  e.  RR  ->  -.  ( _i  x.  A
)  e.  RR+ )
22 df-nel 2456 . 2  |-  ( ( _i  x.  A )  e/  RR+  <->  -.  ( _i  x.  A )  e.  RR+ )
2321, 22sylibr 134 1  |-  ( A  e.  RR  ->  (
_i  x.  A )  e/  RR+ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364    e. wcel 2160    e/ wnel 2455   ` cfv 5235  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842   _ici 7844    + caddc 7845    x. cmul 7847   RR+crp 9685   Recre 10884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-2 9009  df-rp 9686  df-cj 10886  df-re 10887  df-im 10888
This theorem is referenced by: (None)
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