Theorem addcan 8358

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 8357	. . 3 |- ( A e. CC -> E. x e. CC ( x + A ) = 0 )
2	1	3ad2ant1 1044	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> E. x e. CC ( x + A ) = 0 )
3		oveq2 6025	. . . 4 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
4		simprr 533	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + A ) = 0 )
5	4	oveq1d 6032	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( 0 + B ) )
6		simprl 531	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> x e. CC )
7		simpl1 1026	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> A e. CC )
8		simpl2 1027	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> B e. CC )
9	6, 7, 8	addassd 8201	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		addlid 8317	. . . . . . 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 2272	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + B ) ) = B )
13	4	oveq1d 6032	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( 0 + C ) )
14		simpl3 1028	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> C e. CC )
15	6, 7, 14	addassd 8201	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
16		addlid 8317	. . . . . . 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 2272	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + C ) ) = C )
19	12, 18	eqeq12d 2246	. . . 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	imbitrid 154	. . 3 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
21		oveq2 6025	. . 3 |- ( B = C -> ( A + B ) = ( A + C ) )
22	20, 21	impbid1 142	. 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 2655	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511  (class class class)co 6017   CCcc 8029   0cc0 8031   + caddc 8034

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020

This theorem is referenced by:  addcani  8360  addcand  8362  subcan  8433