Theorem intfrac

Description: Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008)

	Ref	Expression
Assertion	intfrac	|- ( A e. RR -> A = ( ( |_ ` A ) + ( A mod 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		modfrac	|- ( A e. RR -> ( A mod 1 ) = ( A - ( |_ ` A ) ) )
2	1	oveq2d	|- ( A e. RR -> ( ( |_ ` A ) + ( A mod 1 ) ) = ( ( |_ ` A ) + ( A - ( |_ ` A ) ) ) )
3		reflcl	|- ( A e. RR -> ( |_ ` A ) e. RR )
4	3	recnd	|- ( A e. RR -> ( |_ ` A ) e. CC )
5		recn	|- ( A e. RR -> A e. CC )
6	4 5	pncan3d	|- ( A e. RR -> ( ( |_ ` A ) + ( A - ( |_ ` A ) ) ) = A )
7	2 6	eqtr2d	|- ( A e. RR -> A = ( ( |_ ` A ) + ( A mod 1 ) ) )