Theorem rennim 11683

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

Step	Hyp	Ref	Expression
1		ax-icn 8221	. . . . . . 7 |- _i e. CC
2		recn 8259	. . . . . . 7 |- ( A e. RR -> A e. CC )
3		mulcl 8253	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
4	1, 2, 3	sylancr 414	. . . . . 6 |- ( A e. RR -> ( _i x. A ) e. CC )
5		rpre 9992	. . . . . . 7 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR )
6		rereb 11544	. . . . . . 7 |- ( ( _i x. A ) e. CC -> ( ( _i x. A ) e. RR <-> ( Re ` ( _i x. A ) ) = ( _i x. A ) ) )
7	5, 6	imbitrid 154	. . . . . 6 |- ( ( _i x. A ) e. CC -> ( ( _i x. A ) e. RR+ -> ( Re ` ( _i x. A ) ) = ( _i x. A ) ) )
8	4, 7	syl 14	. . . . 5 |- ( A e. RR -> ( ( _i x. A ) e. RR+ -> ( Re ` ( _i x. A ) ) = ( _i x. A ) ) )
9	4	addlidd 8422	. . . . . . . 8 |- ( A e. RR -> ( 0 + ( _i x. A ) ) = ( _i x. A ) )
10	9	fveq2d 5673	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = ( Re ` ( _i x. A ) ) )
11		0re 8273	. . . . . . . 8 |- 0 e. RR
12		crre 11538	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( Re ` ( 0 + ( _i x. A ) ) ) = 0 )
13	11, 12	mpan 424	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = 0 )
14	10, 13	eqtr3d 2267	. . . . . 6 |- ( A e. RR -> ( Re ` ( _i x. A ) ) = 0 )
15	14	eqeq1d 2241	. . . . 5 |- ( A e. RR -> ( ( Re ` ( _i x. A ) ) = ( _i x. A ) <-> 0 = ( _i x. A ) ) )
16	8, 15	sylibd 149	. . . 4 |- ( A e. RR -> ( ( _i x. A ) e. RR+ -> 0 = ( _i x. A ) ) )
17		rpne0 10001	. . . . . 6 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) =/= 0 )
18	17	necon2bi 2467	. . . . 5 |- ( ( _i x. A ) = 0 -> -. ( _i x. A ) e. RR+ )
19	18	eqcoms 2235	. . . 4 |- ( 0 = ( _i x. A ) -> -. ( _i x. A ) e. RR+ )
20	16, 19	syl6 33	. . 3 |- ( A e. RR -> ( ( _i x. A ) e. RR+ -> -. ( _i x. A ) e. RR+ ) )
21	20	pm2.01d 623	. 2 |- ( A e. RR -> -. ( _i x. A ) e. RR+ )
22		df-nel 2508	. 2 |- ( ( _i x. A ) e/ RR+ <-> -. ( _i x. A ) e. RR+ )
23	21, 22	sylibr 134	1 |- ( A e. RR -> ( _i x. A ) e/ RR+ )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 2203   e/ wnel 2507   ` cfv 5351  (class class class)co 6049   CCcc 8124   RRcr 8125   0cc0 8126   _ici 8128   + caddc 8129   x. cmul 8131   RR+crp 9985   Recre 11521

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-2 9295  df-rp 9986  df-cj 11523  df-re 11524  df-im 11525

This theorem is referenced by: (None)