Theorem addcan2 8224

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 8221	. . 3 |- ( C e. CC -> E. x e. CC ( C + x ) = 0 )
2	1	3ad2ant3 1022	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. CC ( C + x ) = 0 )
3		oveq1 5932	. . . 4 |- ( ( A + C ) = ( B + C ) -> ( ( A + C ) + x ) = ( ( B + C ) + x ) )
4		simpl1 1002	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> A e. CC )
5		simpl3 1004	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> C e. CC )
6		simprl 529	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> x e. CC )
7	4, 5, 6	addassd 8066	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) + x ) = ( A + ( C + x ) ) )
8		simprr 531	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( C + x ) = 0 )
9	8	oveq2d 5941	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( A + ( C + x ) ) = ( A + 0 ) )
10		addrid 8181	. . . . . . 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 2233	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( A + C ) + x ) = A )
13		simpl2 1003	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> B e. CC )
14	13, 5, 6	addassd 8066	. . . . . 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 5941	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( B + ( C + x ) ) = ( B + 0 ) )
16		addrid 8181	. . . . . . 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 2233	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( C + x ) = 0 ) ) -> ( ( B + C ) + x ) = B )
19	12, 18	eqeq12d 2211	. . . 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 5932	. . 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 2619	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 980   = wceq 1364   e. wcel 2167   E.wrex 2476  (class class class)co 5925   CCcc 7894   0cc0 7896   + caddc 7899

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  addcan2i  8226  addcan2d  8228  muleqadd  8712