Theorem addcan 7966

Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem addcan

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnegex2 7965	. . 3 |- ( A e. CC -> E. x e. CC ( x + A ) = 0 )
2	1	3ad2ant1 1003	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. CC ( x + A ) = 0 )
3		oveq2 5790	. . . 4 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
4		simprr 522	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + A ) = 0 )
5	4	oveq1d 5797	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( 0 + B ) )
6		simprl 521	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> x e. CC )
7		simpl1 985	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> A e. CC )
8		simpl2 986	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> B e. CC )
9	6, 7, 8	addassd 7812	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		addid2 7925	. . . . . . 7 |- ( B e. CC -> ( 0 + B ) = B )
11	8, 10	syl 14	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( 0 + B ) = B )
12	5, 9, 11	3eqtr3d 2181	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + B ) ) = B )
13	4	oveq1d 5797	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( 0 + C ) )
14		simpl3 987	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> C e. CC )
15	6, 7, 14	addassd 7812	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
16		addid2 7925	. . . . . . 7 |- ( C e. CC -> ( 0 + C ) = C )
17	14, 16	syl 14	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( 0 + C ) = C )
18	13, 15, 17	3eqtr3d 2181	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + C ) ) = C )
19	12, 18	eqeq12d 2155	. . . 4 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + ( A + B ) ) = ( x + ( A + C ) ) <-> B = C ) )
20	3, 19	syl5ib 153	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
21		oveq2 5790	. . 3 |- ( B = C -> ( A + B ) = ( A + C ) )
22	20, 21	impbid1 141	. 2 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
23	2, 22	rexlimddv 2557	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

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:  addcani  7968  addcand  7970  subcan  8041