Theorem addcomi 8477

Description: Addition is commutative. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Hypotheses

Ref	Expression
mul.1	|- A e. CC
mul.2	|- B e. CC

Assertion

Ref	Expression
addcomi	|- ( A + B ) = ( B + A )

Proof of Theorem addcomi

Step	Hyp	Ref	Expression
1		mul.1	. 2 |- A e. CC
2		mul.2	. 2 |- B e. CC
3		addcom 8470	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
4	1, 2, 3	mp2an 430	1 |- ( A + B ) = ( B + A )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   + caddc 8188

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addcom 8285

This theorem is used by:  addcomli  8478  add42i  8499  mvlladdi  8551  3m1e2  9433  fztpval  10506  fzo0to42pr  10654  ef01bndlem  12547  modxai  13223  tangtx  16082  log2ublem2  16239  ppiqub  16310  bposlem8  16335  lgsdir2lem2  16370  lgsdir2lem3  16371  lgsdir2lem5  16373