Theorem cnegexlem1 7283

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7286. (Contributed by Eric Schmidt, 22-May-2007.)

Assertion

Ref	Expression
cnegexlem1	|- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Proof of Theorem cnegexlem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 7085	. . . 4 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
2	1	3ad2ant1 959	. . 3 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> E. x e. RR ( A + x ) = 0 )
3		recn 7106	. . . 4 |- ( A e. RR -> A e. CC )
4		recn 7106	. . . . . . 7 |- ( x e. RR -> x e. CC )
5		oveq2 5540	. . . . . . . . . . 11 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
6		simpr 108	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> x e. CC )
7		simpll 495	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> A e. CC )
8		simplrl 501	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> B e. CC )
9	6, 7, 8	addassd 7141	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		simplrr 502	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> C e. CC )
11	6, 7, 10	addassd 7141	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
12	9, 11	eqeq12d 2095	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( x + ( A + B ) ) = ( x + ( A + C ) ) ) )
13	5, 12	syl5ibr 154	. . . . . . . . . 10 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + B ) = ( A + C ) -> ( ( x + A ) + B ) = ( ( x + A ) + C ) ) )
14	13	adantr 270	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( A + x ) = 0 ) -> ( ( A + B ) = ( A + C ) -> ( ( x + A ) + B ) = ( ( x + A ) + C ) ) )
15		addcom 7245	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ x e. CC ) -> ( A + x ) = ( x + A ) )
16	15	eqeq1d 2089	. . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
17	16	adantlr 460	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
18		oveq1 5539	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + B ) = ( 0 + B ) )
19		oveq1 5539	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + C ) = ( 0 + C ) )
20	18, 19	eqeq12d 2095	. . . . . . . . . . . . . 14 |- ( ( x + A ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( 0 + B ) = ( 0 + C ) ) )
21	20	adantl 271	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( x + A ) = 0 ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( 0 + B ) = ( 0 + C ) ) )
22		addid2 7247	. . . . . . . . . . . . . . . 16 |- ( B e. CC -> ( 0 + B ) = B )
23		addid2 7247	. . . . . . . . . . . . . . . 16 |- ( C e. CC -> ( 0 + C ) = C )
24	22, 23	eqeqan12d 2096	. . . . . . . . . . . . . . 15 |- ( ( B e. CC /\ C e. CC ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
25	24	adantl 271	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
26	25	ad2antrr 471	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( x + A ) = 0 ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
27	21, 26	bitrd 186	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( x + A ) = 0 ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) )
28	27	ex 113	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) ) )
29	17, 28	sylbid 148	. . . . . . . . . 10 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) ) )
30	29	imp 122	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( A + x ) = 0 ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) )
31	14, 30	sylibd 147	. . . . . . . 8 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( A + x ) = 0 ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
32	31	ex 113	. . . . . . 7 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
33	4, 32	sylan2 280	. . . . . 6 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. RR ) -> ( ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
34	33	rexlimdva 2477	. . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( E. x e. RR ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
35	34	3impb 1134	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( E. x e. RR ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
36	3, 35	syl3an1 1202	. . 3 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( E. x e. RR ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
37	2, 36	mpd 13	. 2 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
38		oveq2 5540	. 2 |- ( B = C -> ( A + B ) = ( A + C ) )
39	37, 38	impbid1 140	1 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   + caddc 6984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  cnegexlem3  7285