Theorem addcan2 8454

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 8451	. . 3 |- ( C e. CC -> E. x e. CC ( C + x ) = 0 )
2	1	3ad2ant3 1047	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. CC ( C + x ) = 0 )
3		oveq1 6057	. . . 4 |- ( ( A + C ) = ( B + C ) -> ( ( A + C ) + x ) = ( ( B + C ) + x ) )
4		simpl1 1027	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> A e. CC )
5		simpl3 1029	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> C e. CC )
6		simprl 531	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> x e. CC )
7	4, 5, 6	addassd 8296	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) + x ) = ( A + ( C + x ) ) )
8		simprr 533	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( C + x ) = 0 )
9	8	oveq2d 6066	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( A + ( C + x ) ) = ( A + 0 ) )
10		addrid 8411	. . . . . . 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 2269	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) + x ) = A )
13		simpl2 1028	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> B e. CC )
14	13, 5, 6	addassd 8296	. . . . . 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 6066	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( B + ( C + x ) ) = ( B + 0 ) )
16		addrid 8411	. . . . . . 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 2269	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( B + C ) + x ) = B )
19	12, 18	eqeq12d 2247	. . . 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	imbitrid 154	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) = ( B + C ) -> A = B ) )
21		oveq1 6057	. . 3 |- ( A = B -> ( A + C ) = ( B + C ) )
22	20, 21	impbid1 142	. 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 2665	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521  (class class class)co 6050   CCcc 8125   0cc0 8127   + caddc 8130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  addcan2i  8456  addcan2d  8458  muleqadd  8942