Theorem cnegex 10217

Description: Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
cnegex	|- ( A e. CC -> E. x e. CC ( A + x ) = 0 )

Distinct variable group:   x, A

Proof of Theorem cnegex

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 10036	. 2 |- ( A e. CC -> E. a e. RR E. b e. RR A = ( a + ( _i x. b ) ) )
2		ax-rnegex 10007	. . . . . . 7 |- ( a e. RR -> E. c e. RR ( a + c ) = 0 )
3		ax-rnegex 10007	. . . . . . 7 |- ( b e. RR -> E. d e. RR ( b + d ) = 0 )
4	2, 3	anim12i 590	. . . . . 6 |- ( ( a e. RR /\ b e. RR ) -> ( E. c e. RR ( a + c ) = 0 /\ E. d e. RR ( b + d ) = 0 ) )
5		reeanv 3107	. . . . . 6 |- ( E. c e. RR E. d e. RR ( ( a + c ) = 0 /\ ( b + d ) = 0 ) <-> ( E. c e. RR ( a + c ) = 0 /\ E. d e. RR ( b + d ) = 0 ) )
6	4, 5	sylibr 224	. . . . 5 |- ( ( a e. RR /\ b e. RR ) -> E. c e. RR E. d e. RR ( ( a + c ) = 0 /\ ( b + d ) = 0 ) )
7		ax-icn 9995	. . . . . . . . . . 11 |- _i e. CC
8	7	a1i 11	. . . . . . . . . 10 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> _i e. CC )
9		simplrr 801	. . . . . . . . . . 11 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> d e. RR )
10	9	recnd 10068	. . . . . . . . . 10 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> d e. CC )
11	8, 10	mulcld 10060	. . . . . . . . 9 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( _i x. d ) e. CC )
12		simplrl 800	. . . . . . . . . 10 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> c e. RR )
13	12	recnd 10068	. . . . . . . . 9 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> c e. CC )
14	11, 13	addcld 10059	. . . . . . . 8 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( _i x. d ) + c ) e. CC )
15		simplll 798	. . . . . . . . . . . . . 14 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> a e. RR )
16	15	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> a e. CC )
17		simpllr 799	. . . . . . . . . . . . . . 15 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> b e. RR )
18	17	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> b e. CC )
19	8, 18	mulcld 10060	. . . . . . . . . . . . 13 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( _i x. b ) e. CC )
20	16, 19, 11	addassd 10062	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( a + ( _i x. b ) ) + ( _i x. d ) ) = ( a + ( ( _i x. b ) + ( _i x. d ) ) ) )
21	8, 18, 10	adddid 10064	. . . . . . . . . . . . . 14 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( _i x. ( b + d ) ) = ( ( _i x. b ) + ( _i x. d ) ) )
22		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( b + d ) = 0 )
23	22	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( _i x. ( b + d ) ) = ( _i x. 0 ) )
24		mul01 10215	. . . . . . . . . . . . . . . 16 |- ( _i e. CC -> ( _i x. 0 ) = 0 )
25	7, 24	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( _i x. 0 ) = 0
26	23, 25	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( _i x. ( b + d ) ) = 0 )
27	21, 26	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( _i x. b ) + ( _i x. d ) ) = 0 )
28	27	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( a + ( ( _i x. b ) + ( _i x. d ) ) ) = ( a + 0 ) )
29		addid1 10216	. . . . . . . . . . . . 13 |- ( a e. CC -> ( a + 0 ) = a )
30	16, 29	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( a + 0 ) = a )
31	20, 28, 30	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( a + ( _i x. b ) ) + ( _i x. d ) ) = a )
32	31	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( ( a + ( _i x. b ) ) + ( _i x. d ) ) + c ) = ( a + c ) )
33	16, 19	addcld 10059	. . . . . . . . . . 11 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( a + ( _i x. b ) ) e. CC )
34	33, 11, 13	addassd 10062	. . . . . . . . . 10 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( ( a + ( _i x. b ) ) + ( _i x. d ) ) + c ) = ( ( a + ( _i x. b ) ) + ( ( _i x. d ) + c ) ) )
35	32, 34	eqtr3d 2658	. . . . . . . . 9 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( a + c ) = ( ( a + ( _i x. b ) ) + ( ( _i x. d ) + c ) ) )
36		simprl 794	. . . . . . . . 9 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( a + c ) = 0 )
37	35, 36	eqtr3d 2658	. . . . . . . 8 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> ( ( a + ( _i x. b ) ) + ( ( _i x. d ) + c ) ) = 0 )
38		oveq2 6658	. . . . . . . . . 10 |- ( x = ( ( _i x. d ) + c ) -> ( ( a + ( _i x. b ) ) + x ) = ( ( a + ( _i x. b ) ) + ( ( _i x. d ) + c ) ) )
39	38	eqeq1d 2624	. . . . . . . . 9 |- ( x = ( ( _i x. d ) + c ) -> ( ( ( a + ( _i x. b ) ) + x ) = 0 <-> ( ( a + ( _i x. b ) ) + ( ( _i x. d ) + c ) ) = 0 ) )
40	39	rspcev 3309	. . . . . . . 8 |- ( ( ( ( _i x. d ) + c ) e. CC /\ ( ( a + ( _i x. b ) ) + ( ( _i x. d ) + c ) ) = 0 ) -> E. x e. CC ( ( a + ( _i x. b ) ) + x ) = 0 )
41	14, 37, 40	syl2anc 693	. . . . . . 7 |- ( ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) /\ ( ( a + c ) = 0 /\ ( b + d ) = 0 ) ) -> E. x e. CC ( ( a + ( _i x. b ) ) + x ) = 0 )
42	41	ex 450	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ ( c e. RR /\ d e. RR ) ) -> ( ( ( a + c ) = 0 /\ ( b + d ) = 0 ) -> E. x e. CC ( ( a + ( _i x. b ) ) + x ) = 0 ) )
43	42	rexlimdvva 3038	. . . . 5 |- ( ( a e. RR /\ b e. RR ) -> ( E. c e. RR E. d e. RR ( ( a + c ) = 0 /\ ( b + d ) = 0 ) -> E. x e. CC ( ( a + ( _i x. b ) ) + x ) = 0 ) )
44	6, 43	mpd 15	. . . 4 |- ( ( a e. RR /\ b e. RR ) -> E. x e. CC ( ( a + ( _i x. b ) ) + x ) = 0 )
45		oveq1 6657	. . . . . 6 |- ( A = ( a + ( _i x. b ) ) -> ( A + x ) = ( ( a + ( _i x. b ) ) + x ) )
46	45	eqeq1d 2624	. . . . 5 |- ( A = ( a + ( _i x. b ) ) -> ( ( A + x ) = 0 <-> ( ( a + ( _i x. b ) ) + x ) = 0 ) )
47	46	rexbidv 3052	. . . 4 |- ( A = ( a + ( _i x. b ) ) -> ( E. x e. CC ( A + x ) = 0 <-> E. x e. CC ( ( a + ( _i x. b ) ) + x ) = 0 ) )
48	44, 47	syl5ibrcom 237	. . 3 |- ( ( a e. RR /\ b e. RR ) -> ( A = ( a + ( _i x. b ) ) -> E. x e. CC ( A + x ) = 0 ) )
49	48	rexlimivv 3036	. 2 |- ( E. a e. RR E. b e. RR A = ( a + ( _i x. b ) ) -> E. x e. CC ( A + x ) = 0 )
50	1, 49	syl 17	1 |- ( A e. CC -> E. x e. CC ( A + x ) = 0 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   _ici 9938   + caddc 9939   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079

This theorem is referenced by:  addid2  10219  addcan2  10221  0cnALT  10270  negeu  10271