Theorem rennim 11712

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 8238	. . . . . . 7 |- _i e. CC
2		recn 8276	. . . . . . 7 |- ( A e. RR -> A e. CC )
3		mulcl 8270	. . . . . . 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 10011	. . . . . . 7 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR )
6		rereb 11573	. . . . . . 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 8439	. . . . . . . 8 |- ( A e. RR -> ( 0 + ( _i x. A ) ) = ( _i x. A ) )
10	9	fveq2d 5679	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = ( Re ` ( _i x. A ) ) )
11		0re 8290	. . . . . . . 8 |- 0 e. RR
12		crre 11567	. . . . . . . 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 2269	. . . . . 6 |- ( A e. RR -> ( Re ` ( _i x. A ) ) = 0 )
15	14	eqeq1d 2243	. . . . 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 10020	. . . . . 6 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) =/= 0 )
18	17	necon2bi 2469	. . . . 5 |- ( ( _i x. A ) = 0 -> -. ( _i x. A ) e. RR+ )
19	18	eqcoms 2237	. . . 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 2510	. 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 2205   e/ wnel 2509   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   _ici 8145   + caddc 8146   x. cmul 8148   RR+crp 10004   Recre 11550

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-2 9313  df-rp 10005  df-cj 11552  df-re 11553  df-im 11554

This theorem is referenced by: (None)