Theorem rennim 10966

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 7869	. . . . . . 7 |- _i e. CC
2		recn 7907	. . . . . . 7 |- ( A e. RR -> A e. CC )
3		mulcl 7901	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
4	1, 2, 3	sylancr 412	. . . . . 6 |- ( A e. RR -> ( _i x. A ) e. CC )
5		rpre 9617	. . . . . . 7 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) e. RR )
6		rereb 10827	. . . . . . 7 |- ( ( _i x. A ) e. CC -> ( ( _i x. A ) e. RR <-> ( Re ` ( _i x. A ) ) = ( _i x. A ) ) )
7	5, 6	syl5ib 153	. . . . . 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	addid2d 8069	. . . . . . . 8 |- ( A e. RR -> ( 0 + ( _i x. A ) ) = ( _i x. A ) )
10	9	fveq2d 5500	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = ( Re ` ( _i x. A ) ) )
11		0re 7920	. . . . . . . 8 |- 0 e. RR
12		crre 10821	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( Re ` ( 0 + ( _i x. A ) ) ) = 0 )
13	11, 12	mpan 422	. . . . . . 7 |- ( A e. RR -> ( Re ` ( 0 + ( _i x. A ) ) ) = 0 )
14	10, 13	eqtr3d 2205	. . . . . 6 |- ( A e. RR -> ( Re ` ( _i x. A ) ) = 0 )
15	14	eqeq1d 2179	. . . . 5 |- ( A e. RR -> ( ( Re ` ( _i x. A ) ) = ( _i x. A ) <-> 0 = ( _i x. A ) ) )
16	8, 15	sylibd 148	. . . 4 |- ( A e. RR -> ( ( _i x. A ) e. RR+ -> 0 = ( _i x. A ) ) )
17		rpne0 9626	. . . . . 6 |- ( ( _i x. A ) e. RR+ -> ( _i x. A ) =/= 0 )
18	17	necon2bi 2395	. . . . 5 |- ( ( _i x. A ) = 0 -> -. ( _i x. A ) e. RR+ )
19	18	eqcoms 2173	. . . 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 613	. 2 |- ( A e. RR -> -. ( _i x. A ) e. RR+ )
22		df-nel 2436	. 2 |- ( ( _i x. A ) e/ RR+ <-> -. ( _i x. A ) e. RR+ )
23	21, 22	sylibr 133	1 |- ( A e. RR -> ( _i x. A ) e/ RR+ )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   e. wcel 2141   e/ wnel 2435   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   _ici 7776   + caddc 7777   x. cmul 7779   RR+crp 9610   Recre 10804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-2 8937  df-rp 9611  df-cj 10806  df-re 10807  df-im 10808

This theorem is referenced by: (None)