Theorem rennim 11770

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 8274	. . . . . . 7 |- _i e. CC
2		recn 8312	. . . . . . 7 |- ( A e. RR -> A e. CC )
3		mulcl 8306	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
4	1, 2, 3	sylancr 418	. . . . . 6 |- ( A e. RR -> ( _i x. A ) e. CC )
5		rpre 10063	. . . . . . 7 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR )
6		rereb 11630	. . . . . . 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 8476	. . . . . . . 8 |- ( A e. RR -> ( 0 + ( _i x. A ) ) = ( _i x. A ) )
10	9	fveq2d 5699	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = ( Re ` ( _i x. A ) ) )
11		0re 8326	. . . . . . . 8 |- 0 e. RR
12		crre 11624	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( Re ` ( 0 + ( _i x. A ) ) ) = 0 )
13	11, 12	mpan 428	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = 0 )
14	10, 13	eqtr3d 2273	. . . . . 6 |- ( A e. RR -> ( Re ` ( _i x. A ) ) = 0 )
15	14	eqeq1d 2247	. . . . 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 10072	. . . . . 6 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) =/= 0 )
18	17	necon2bi 2475	. . . . 5 |- ( ( _i x. A ) = 0 -> -. ( _i x. A ) e. RR+ )
19	18	eqcoms 2241	. . . 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 627	. 2 |- ( A e. RR -> -. ( _i x. A ) e. RR+ )
22		df-nel 2516	. 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+ )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   e/ wnel 2515   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   _ici 8181   + caddc 8182   x. cmul 8184   RR+crp 10056   Recre 11607

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-2 9364  df-rp 10057  df-cj 11609  df-re 11610  df-im 11611

This theorem is used by: (None)