Theorem cnegexlem1 8073

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8076. (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 7862	. . . 4 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
2	1	3ad2ant1 1008	. . 3 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> E. x e. RR ( A + x ) = 0 )
3		recn 7886	. . . 4 |- ( A e. RR -> A e. CC )
4		recn 7886	. . . . . . 7 |- ( x e. RR -> x e. CC )
5		oveq2 5850	. . . . . . . . . . 11 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
6		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> x e. CC )
7		simpll 519	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> A e. CC )
8		simplrl 525	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> B e. CC )
9	6, 7, 8	addassd 7921	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		simplrr 526	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> C e. CC )
11	6, 7, 10	addassd 7921	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
12	9, 11	eqeq12d 2180	. . . . . . . . . . 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 155	. . . . . . . . . 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 274	. . . . . . . . 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 8035	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ x e. CC ) -> ( A + x ) = ( x + A ) )
16	15	eqeq1d 2174	. . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
17	16	adantlr 469	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
18		oveq1 5849	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + B ) = ( 0 + B ) )
19		oveq1 5849	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + C ) = ( 0 + C ) )
20	18, 19	eqeq12d 2180	. . . . . . . . . . . . . 14 |- ( ( x + A ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( 0 + B ) = ( 0 + C ) ) )
21	20	adantl 275	. . . . . . . . . . . . 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 8037	. . . . . . . . . . . . . . . 16 |- ( B e. CC -> ( 0 + B ) = B )
23		addid2 8037	. . . . . . . . . . . . . . . 16 |- ( C e. CC -> ( 0 + C ) = C )
24	22, 23	eqeqan12d 2181	. . . . . . . . . . . . . . 15 |- ( ( B e. CC /\ C e. CC ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
25	24	adantl 275	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
26	25	ad2antrr 480	. . . . . . . . . . . . 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 187	. . . . . . . . . . . 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 114	. . . . . . . . . . 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 149	. . . . . . . . . 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 123	. . . . . . . . 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 148	. . . . . . . 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 114	. . . . . . 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 284	. . . . . 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 2583	. . . . 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 1189	. . . 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 1261	. . 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 5850	. 2 |- ( B = C -> ( A + B ) = ( A + C ) )
39	37, 38	impbid1 141	1 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   + caddc 7756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  cnegexlem3  8075