Theorem cnegexlem2 7719

Description: Existence of a real number which produces a real number when multiplied by _i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 7721. (Contributed by Eric Schmidt, 22-May-2007.)

Assertion

Ref	Expression
cnegexlem2	|- E. y e. RR ( _i x. y ) e. RR

Proof of Theorem cnegexlem2

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 7541	. 2 |- 0 e. CC
2		cnre 7545	. 2 |- ( 0 e. CC -> E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) )
3		ax-rnegex 7515	. . . . . 6 |- ( x e. RR -> E. z e. RR ( x + z ) = 0 )
4	3	adantr 271	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> E. z e. RR ( x + z ) = 0 )
5		recn 7536	. . . . . . . . . . 11 |- ( x e. RR -> x e. CC )
6		ax-icn 7501	. . . . . . . . . . . 12 |- _i e. CC
7		recn 7536	. . . . . . . . . . . 12 |- ( y e. RR -> y e. CC )
8		mulcl 7530	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
9	6, 7, 8	sylancr 406	. . . . . . . . . . 11 |- ( y e. RR -> ( _i x. y ) e. CC )
10		recn 7536	. . . . . . . . . . 11 |- ( z e. RR -> z e. CC )
11		addid2 7682	. . . . . . . . . . . . . . 15 |- ( z e. CC -> ( 0 + z ) = z )
12	11	3ad2ant3 967	. . . . . . . . . . . . . 14 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( 0 + z ) = z )
13	12	adantr 271	. . . . . . . . . . . . 13 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> ( 0 + z ) = z )
14		oveq1 5673	. . . . . . . . . . . . . . 15 |- ( ( x + z ) = 0 -> ( ( x + z ) + ( _i x. y ) ) = ( 0 + ( _i x. y ) ) )
15	14	ad2antrl 475	. . . . . . . . . . . . . 14 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> ( ( x + z ) + ( _i x. y ) ) = ( 0 + ( _i x. y ) ) )
16		add32 7702	. . . . . . . . . . . . . . . . 17 |- ( ( x e. CC /\ z e. CC /\ ( _i x. y ) e. CC ) -> ( ( x + z ) + ( _i x. y ) ) = ( ( x + ( _i x. y ) ) + z ) )
17	16	3com23 1150	. . . . . . . . . . . . . . . 16 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( ( x + z ) + ( _i x. y ) ) = ( ( x + ( _i x. y ) ) + z ) )
18		oveq1 5673	. . . . . . . . . . . . . . . . 17 |- ( 0 = ( x + ( _i x. y ) ) -> ( 0 + z ) = ( ( x + ( _i x. y ) ) + z ) )
19	18	eqcomd 2094	. . . . . . . . . . . . . . . 16 |- ( 0 = ( x + ( _i x. y ) ) -> ( ( x + ( _i x. y ) ) + z ) = ( 0 + z ) )
20	17, 19	sylan9eq 2141	. . . . . . . . . . . . . . 15 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ 0 = ( x + ( _i x. y ) ) ) -> ( ( x + z ) + ( _i x. y ) ) = ( 0 + z ) )
21	20	adantrl 463	. . . . . . . . . . . . . 14 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> ( ( x + z ) + ( _i x. y ) ) = ( 0 + z ) )
22		addid2 7682	. . . . . . . . . . . . . . . 16 |- ( ( _i x. y ) e. CC -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
23	22	3ad2ant2 966	. . . . . . . . . . . . . . 15 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
24	23	adantr 271	. . . . . . . . . . . . . 14 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
25	15, 21, 24	3eqtr3d 2129	. . . . . . . . . . . . 13 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> ( 0 + z ) = ( _i x. y ) )
26	13, 25	eqtr3d 2123	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> z = ( _i x. y ) )
27	26	ex 114	. . . . . . . . . . 11 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) -> z = ( _i x. y ) ) )
28	5, 9, 10, 27	syl3an 1217	. . . . . . . . . 10 |- ( ( x e. RR /\ y e. RR /\ z e. RR ) -> ( ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) -> z = ( _i x. y ) ) )
29	28	3expa 1144	. . . . . . . . 9 |- ( ( ( x e. RR /\ y e. RR ) /\ z e. RR ) -> ( ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) -> z = ( _i x. y ) ) )
30	29	imp 123	. . . . . . . 8 |- ( ( ( ( x e. RR /\ y e. RR ) /\ z e. RR ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> z = ( _i x. y ) )
31		simplr 498	. . . . . . . 8 |- ( ( ( ( x e. RR /\ y e. RR ) /\ z e. RR ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> z e. RR )
32	30, 31	eqeltrrd 2166	. . . . . . 7 |- ( ( ( ( x e. RR /\ y e. RR ) /\ z e. RR ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> ( _i x. y ) e. RR )
33	32	exp32 358	. . . . . 6 |- ( ( ( x e. RR /\ y e. RR ) /\ z e. RR ) -> ( ( x + z ) = 0 -> ( 0 = ( x + ( _i x. y ) ) -> ( _i x. y ) e. RR ) ) )
34	33	rexlimdva 2490	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( E. z e. RR ( x + z ) = 0 -> ( 0 = ( x + ( _i x. y ) ) -> ( _i x. y ) e. RR ) ) )
35	4, 34	mpd 13	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( 0 = ( x + ( _i x. y ) ) -> ( _i x. y ) e. RR ) )
36	35	reximdva 2476	. . 3 |- ( x e. RR -> ( E. y e. RR 0 = ( x + ( _i x. y ) ) -> E. y e. RR ( _i x. y ) e. RR ) )
37	36	rexlimiv 2484	. 2 |- ( E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) -> E. y e. RR ( _i x. y ) e. RR )
38	1, 2, 37	mp2b 8	1 |- E. y e. RR ( _i x. y ) e. RR

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 925   = wceq 1290   e. wcel 1439   E.wrex 2361  (class class class)co 5666   CCcc 7409   RRcr 7410   0cc0 7411   _ici 7413   + caddc 7414   x. cmul 7416

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7498  ax-1cn 7499  ax-icn 7501  ax-addcl 7502  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-i2m1 7511  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669

This theorem is referenced by:  cnegex  7721