Theorem addcani 8361

Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Hypotheses

Ref	Expression
addcani.1	|- A e. CC
addcani.2	|- B e. CC
addcani.3	|- C e. CC

Assertion

Ref	Expression
addcani	|- ( ( A + B ) = ( A + C ) <-> B = C )

Proof of Theorem addcani

Step	Hyp	Ref	Expression
1		addcani.1	. 2 |- A e. CC
2		addcani.2	. 2 |- B e. CC
3		addcani.3	. 2 |- C e. CC
4		addcan 8359	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
5	1, 2, 3, 4	mp3an 1373	1 |- ( ( A + B ) = ( A + C ) <-> B = C )

Syntax hints:   <-> wb 105   = wceq 1397   e. wcel 2202  (class class class)co 6018   CCcc 8030   + caddc 8035

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 8124  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

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 6021

This theorem is referenced by:  negdii  8463  fsumrelem  12033  karatsuba  13004