Theorem flmod

Description: The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008)

	Ref	Expression
Assertion	flmod	|- ( 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		recn	|- ( A e. RR -> A e. CC )
4		reflcl	|- ( A e. RR -> ( |_ ` A ) e. RR )
5	4	recnd	|- ( A e. RR -> ( |_ ` A ) e. CC )
6	3 5	nncand	|- ( A e. RR -> ( A - ( A - ( |_ ` A ) ) ) = ( |_ ` A ) )
7	2 6	eqtr2d	|- ( A e. RR -> ( |_ ` A ) = ( A - ( A mod 1 ) ) )