Theorem cnegexlem2 7938

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 7940. (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 7758	. 2 |- 0 e. CC
2		cnre 7762	. 2 |- ( 0 e. CC -> E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) )
3		ax-rnegex 7729	. . . . . 6 |- ( x e. RR -> E. z e. RR ( x + z ) = 0 )
4	3	adantr 274	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> E. z e. RR ( x + z ) = 0 )
5		recn 7753	. . . . . . . . . . 11 |- ( x e. RR -> x e. CC )
6		ax-icn 7715	. . . . . . . . . . . 12 |- _i e. CC
7		recn 7753	. . . . . . . . . . . 12 |- ( y e. RR -> y e. CC )
8		mulcl 7747	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
9	6, 7, 8	sylancr 410	. . . . . . . . . . 11 |- ( y e. RR -> ( _i x. y ) e. CC )
10		recn 7753	. . . . . . . . . . 11 |- ( z e. RR -> z e. CC )
11		addid2 7901	. . . . . . . . . . . . . . 15 |- ( z e. CC -> ( 0 + z ) = z )
12	11	3ad2ant3 1004	. . . . . . . . . . . . . 14 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( 0 + z ) = z )
13	12	adantr 274	. . . . . . . . . . . . 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 5781	. . . . . . . . . . . . . . 15 |- ( ( x + z ) = 0 -> ( ( x + z ) + ( _i x. y ) ) = ( 0 + ( _i x. y ) ) )
15	14	ad2antrl 481	. . . . . . . . . . . . . 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 7921	. . . . . . . . . . . . . . . . 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 1187	. . . . . . . . . . . . . . . 16 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( ( x + z ) + ( _i x. y ) ) = ( ( x + ( _i x. y ) ) + z ) )
18		oveq1 5781	. . . . . . . . . . . . . . . . 17 |- ( 0 = ( x + ( _i x. y ) ) -> ( 0 + z ) = ( ( x + ( _i x. y ) ) + z ) )
19	18	eqcomd 2145	. . . . . . . . . . . . . . . 16 |- ( 0 = ( x + ( _i x. y ) ) -> ( ( x + ( _i x. y ) ) + z ) = ( 0 + z ) )
20	17, 19	sylan9eq 2192	. . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . 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 7901	. . . . . . . . . . . . . . . 16 |- ( ( _i x. y ) e. CC -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
23	22	3ad2ant2 1003	. . . . . . . . . . . . . . 15 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
24	23	adantr 274	. . . . . . . . . . . . . 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 2180	. . . . . . . . . . . . 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 2174	. . . . . . . . . . . 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 1258	. . . . . . . . . 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 1181	. . . . . . . . 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 519	. . . . . . . 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 2217	. . . . . . 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 362	. . . . . 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 2549	. . . . 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 2534	. . 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 2543	. 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 962   = wceq 1331   e. wcel 1480   E.wrex 2417  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   _ici 7622   + caddc 7623   x. cmul 7625

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  cnegex  7940