Theorem addcan 8034

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 8033	. . 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 5822	. . . 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 5829	. . . . . 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 7879	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		addid2 7993	. . . . . . 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 2195	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + B ) ) = B )
13	4	oveq1d 5829	. . . . . 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 7879	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
16		addid2 7993	. . . . . . 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 2195	. . . . 5 |- ( ( ( A e. CC /\ B e. CC /\ C e. CC ) /\ ( x e. CC /\ ( x + A ) = 0 ) ) -> ( x + ( A + C ) ) = C )
19	12, 18	eqeq12d 2169	. . . 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 5822	. . 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 2576	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 2125   E.wrex 2433  (class class class)co 5814   CCcc 7709   0cc0 7711   + caddc 7714

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136  ax-resscn 7803  ax-1cn 7804  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-iota 5128  df-fv 5171  df-ov 5817

This theorem is referenced by:  addcani  8036  addcand  8038  subcan  8109