Theorem cnegexlem2 6984

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 6986. (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 6817	. 2 |- 0 e. CC
2		cnre 6821	. 2 |- ( 0 e. CC -> E.x e. RR E.y e. RR 0 = (x + ( _i x. y)))
3		ax-rnegex 6792	. . . . . 6 |- (x e. RR -> E.z e. RR (x + z) = 0)
4	3	adantr 261	. . . . 5 |- ((x e. RR /\ y e. RR) -> E.z e. RR (x + z) = 0)
5		recn 6812	. . . . . . . . . . 11 |- (x e. RR -> x e. CC)
6		ax-icn 6778	. . . . . . . . . . . 12 |- _i e. CC
7		recn 6812	. . . . . . . . . . . 12 |- (y e. RR -> y e. CC)
8		mulcl 6806	. . . . . . . . . . . 12 |- (( _i e. CC /\ y e. CC) -> ( _i x. y) e. CC)
9	6, 7, 8	sylancr 393	. . . . . . . . . . 11 |- (y e. RR -> ( _i x. y) e. CC)
10		recn 6812	. . . . . . . . . . 11 |- (z e. RR -> z e. CC)
11		addid2 6949	. . . . . . . . . . . . . . 15 |- (z e. CC -> ( 0 + z) = z)
12	11	3ad2ant3 926	. . . . . . . . . . . . . 14 |- ((x e. CC /\ ( _i x. y) e. CC /\ z e. CC) -> ( 0 + z) = z)
13	12	adantr 261	. . . . . . . . . . . . 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 5462	. . . . . . . . . . . . . . 15 |- ((x + z) = 0 -> ((x + z) + ( _i x. y)) = ( 0 + ( _i x. y)))
15	14	ad2antrl 459	. . . . . . . . . . . . . 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 6967	. . . . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . . . 16 |- ((x e. CC /\ ( _i x. y) e. CC /\ z e. CC) -> ((x + z) + ( _i x. y)) = ((x + ( _i x. y)) + z))
18		oveq1 5462	. . . . . . . . . . . . . . . . 17 |- ( 0 = (x + ( _i x. y)) -> ( 0 + z) = ((x + ( _i x. y)) + z))
19	18	eqcomd 2042	. . . . . . . . . . . . . . . 16 |- ( 0 = (x + ( _i x. y)) -> ((x + ( _i x. y)) + z) = ( 0 + z))
20	17, 19	sylan9eq 2089	. . . . . . . . . . . . . . 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 447	. . . . . . . . . . . . . 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 6949	. . . . . . . . . . . . . . . 16 |- (( _i x. y) e. CC -> ( 0 + ( _i x. y)) = ( _i x. y))
23	22	3ad2ant2 925	. . . . . . . . . . . . . . 15 |- ((x e. CC /\ ( _i x. y) e. CC /\ z e. CC) -> ( 0 + ( _i x. y)) = ( _i x. y))
24	23	adantr 261	. . . . . . . . . . . . . 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 2077	. . . . . . . . . . . . 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 2071	. . . . . . . . . . . 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 108	. . . . . . . . . . 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 1176	. . . . . . . . . 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 1103	. . . . . . . . 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 115	. . . . . . . 8 |- ((((x e. RR /\ y e. RR) /\ z e. RR) /\ ((x + z) = 0 /\ 0 = (x + ( _i x. y)))) -> z = ( _i x. y))
31		simplr 482	. . . . . . . 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 2112	. . . . . . 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 347	. . . . . 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 2427	. . . . 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 2415	. . 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 2421	. 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 97   /\ w3a 884   = wceq 1242   e. wcel 1390  E.wrex 2301  (class class class)co 5455   CCcc 6709   RRcr 6710   0cc0 6711   _ici 6713   + caddc 6714   x. cmul 6716

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-resscn 6775  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458

This theorem is referenced by:  cnegex  6986