Theorem cnegexlem2 8449

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 8451. (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 8266	. 2 |- 0 e. CC
2		cnre 8270	. 2 |- ( 0 e. CC -> E. x e. RR E. y e. RR 0 = ( x + ( _i x. y ) ) )
3		ax-rnegex 8236	. . . . . 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 8260	. . . . . . . . . . 11 |- ( x e. RR -> x e. CC )
6		ax-icn 8222	. . . . . . . . . . . 12 |- _i e. CC
7		recn 8260	. . . . . . . . . . . 12 |- ( y e. RR -> y e. CC )
8		mulcl 8254	. . . . . . . . . . . 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 8260	. . . . . . . . . . 11 |- ( z e. RR -> z e. CC )
11		addlid 8412	. . . . . . . . . . . . . . 15 |- ( z e. CC -> ( 0 + z ) = z )
12	11	3ad2ant3 1047	. . . . . . . . . . . . . 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 6057	. . . . . . . . . . . . . . 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 8432	. . . . . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . . . . 16 |- ( ( x e. CC /\ ( _i x. y ) e. CC /\ z e. CC ) -> ( ( x + z ) + ( _i x. y ) ) = ( ( x + ( _i x. y ) ) + z ) )
18		oveq1 6057	. . . . . . . . . . . . . . . . 17 |- ( 0 = ( x + ( _i x. y ) ) -> ( 0 + z ) = ( ( x + ( _i x. y ) ) + z ) )
19	18	eqcomd 2238	. . . . . . . . . . . . . . . 16 |- ( 0 = ( x + ( _i x. y ) ) -> ( ( x + ( _i x. y ) ) + z ) = ( 0 + z ) )
20	17, 19	sylan9eq 2285	. . . . . . . . . . . . . . 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		addlid 8412	. . . . . . . . . . . . . . . 16 |- ( ( _i x. y ) e. CC -> ( 0 + ( _i x. y ) ) = ( _i x. y ) )
23	22	3ad2ant2 1046	. . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . 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 2267	. . . . . . . . . . . 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 1316	. . . . . . . . . 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 1230	. . . . . . . . 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 529	. . . . . . . 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 2310	. . . . . . 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 2660	. . . . 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 2644	. . 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 2654	. 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   _ici 8129   + caddc 8130   x. cmul 8132

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  cnegex  8451