Theorem cnegexlem2 8123

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 8125. (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 7940	. 2 |- 0 e. CC
2		cnre 7944	. 2 |- ( 0 e. CC -> E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) )
3		ax-rnegex 7911	. . . . . 6 |- ( x e. RR -> E. z e. RR ( x + z ) = 0 )
4	3	adantr 276	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> E. z e. RR ( x + z ) = 0 )
5		recn 7935	. . . . . . . . . . 11 |- ( x e. RR -> x e. CC )
6		ax-icn 7897	. . . . . . . . . . . 12 |- _i e. CC
7		recn 7935	. . . . . . . . . . . 12 |- ( y e. RR -> y e. CC )
8		mulcl 7929	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) e. CC )
9	6, 7, 8	sylancr 414	. . . . . . . . . . 11 |- ( y e. RR -> ( _i x. y ) e. CC )
10		recn 7935	. . . . . . . . . . 11 |- ( z e. RR -> z e. CC )
11		addid2 8086	. . . . . . . . . . . . . . 15 |- ( z e. CC -> ( 0 + z ) = z )
12	11	3ad2ant3 1020	. . . . . . . . . . . . . 14 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( 0 + z ) = z )
13	12	adantr 276	. . . . . . . . . . . . 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 5876	. . . . . . . . . . . . . . 15 |- ( ( x + z ) = 0 -> ( ( x + z ) + ( _i x. y ) ) = ( 0 + ( _i x. y ) ) )
15	14	ad2antrl 490	. . . . . . . . . . . . . 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 8106	. . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . 16 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( ( x + z ) + ( _i x. y ) ) = ( ( x + ( _i x. y ) ) + z ) )
18		oveq1 5876	. . . . . . . . . . . . . . . . 17 |- ( 0 = ( x + ( _i x. y ) ) -> ( 0 + z ) = ( ( x + ( _i x. y ) ) + z ) )
19	18	eqcomd 2183	. . . . . . . . . . . . . . . 16 |- ( 0 = ( x + ( _i x. y ) ) -> ( ( x + ( _i x. y ) ) + z ) = ( 0 + z ) )
20	17, 19	sylan9eq 2230	. . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . . 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 8086	. . . . . . . . . . . . . . . 16 |- ( ( _i x. y ) e. CC -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
23	22	3ad2ant2 1019	. . . . . . . . . . . . . . 15 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
24	23	adantr 276	. . . . . . . . . . . . . 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 2218	. . . . . . . . . . . . 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 2212	. . . . . . . . . . . 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 115	. . . . . . . . . . 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 1280	. . . . . . . . . 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 1203	. . . . . . . . 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 124	. . . . . . . 8 |- ( ( ( ( x e. RR /\ y e. RR ) /\ z e. RR ) /\ ( ( x + z ) = 0 /\ 0 = ( x + ( _i x. y ) ) ) ) -> z = ( _i x. y ) )
31		simplr 528	. . . . . . . 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 2255	. . . . . . 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 365	. . . . . 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 2594	. . . . 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 2579	. . 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 2588	. 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 104   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   _ici 7804   + caddc 7805   x. cmul 7807

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872

This theorem is referenced by:  cnegex  8125