Theorem addcan 7660

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 7659	. . 3 |- ( A e. CC -> E. x e. CC ( x + A ) = 0 )
2	1	3ad2ant1 964	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. CC ( x + A ) = 0 )
3		oveq2 5660	. . . 4 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
4		simprr 499	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + A ) = 0 )
5	4	oveq1d 5667	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( 0 + B ) )
6		simprl 498	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> x e. CC )
7		simpl1 946	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> A e. CC )
8		simpl2 947	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> B e. CC )
9	6, 7, 8	addassd 7508	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		addid2 7619	. . . . . . 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 2128	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + B ) ) = B )
13	4	oveq1d 5667	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( 0 + C ) )
14		simpl3 948	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> C e. CC )
15	6, 7, 14	addassd 7508	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
16		addid2 7619	. . . . . . 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 2128	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + C ) ) = C )
19	12, 18	eqeq12d 2102	. . . 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 152	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
21		oveq2 5660	. . 3 |- ( B = C -> ( A + B ) = ( A + C ) )
22	20, 21	impbid1 140	. 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 2493	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   = wceq 1289   e. wcel 1438   E.wrex 2360  (class class class)co 5652   CCcc 7346   0cc0 7348   + caddc 7351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655

This theorem is referenced by:  addcani  7662  addcand  7664  subcan  7735