Theorem addcan2 7967

Description: Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
addcan2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )

Proof of Theorem addcan2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnegex 7964	. . 3 |- ( C e. CC -> E. x e. CC ( C + x ) = 0 )
2	1	3ad2ant3 1005	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. CC ( C + x ) = 0 )
3		oveq1 5789	. . . 4 |- ( ( A + C ) = ( B + C ) -> ( ( A + C ) + x ) = ( ( B + C ) + x ) )
4		simpl1 985	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> A e. CC )
5		simpl3 987	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> C e. CC )
6		simprl 521	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> x e. CC )
7	4, 5, 6	addassd 7812	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) + x ) = ( A + ( C + x ) ) )
8		simprr 522	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( C + x ) = 0 )
9	8	oveq2d 5798	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( A + ( C + x ) ) = ( A + 0 ) )
10		addid1 7924	. . . . . . 7 |- ( A e. CC -> ( A + 0 ) = A )
11	4, 10	syl 14	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( A + 0 ) = A )
12	7, 9, 11	3eqtrd 2177	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) + x ) = A )
13		simpl2 986	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> B e. CC )
14	13, 5, 6	addassd 7812	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( B + C ) + x ) = ( B + ( C + x ) ) )
15	8	oveq2d 5798	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( B + ( C + x ) ) = ( B + 0 ) )
16		addid1 7924	. . . . . . 7 |- ( B e. CC -> ( B + 0 ) = B )
17	13, 16	syl 14	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( B + 0 ) = B )
18	14, 15, 17	3eqtrd 2177	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( B + C ) + x ) = B )
19	12, 18	eqeq12d 2155	. . . 4 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( ( A + C ) + x ) = ( ( B + C ) + x ) <-> A = B ) )
20	3, 19	syl5ib 153	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) = ( B + C ) -> A = B ) )
21		oveq1 5789	. . 3 |- ( A = B -> ( A + C ) = ( B + C ) )
22	20, 21	impbid1 141	. 2 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
23	2, 22	rexlimddv 2557	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   E.wrex 2418  (class class class)co 5782   CCcc 7642   0cc0 7644   + caddc 7647

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-resscn 7736  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  addcan2i  7969  addcan2d  7971  muleqadd  8453