Theorem fldivle

Description: The floor function of a division of a real number by a positive real number is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018)

	Ref	Expression
Assertion	fldivle	|- ( ( K e. RR /\ L e. RR+ ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )

Proof

Step	Hyp	Ref	Expression
1		rerpdivcl	|- ( ( K e. RR /\ L e. RR+ ) -> ( K / L ) e. RR )
2		fllelt	|- ( ( K / L ) e. RR -> ( ( |_ ` ( K / L ) ) <_ ( K / L ) /\ ( K / L ) < ( ( |_ ` ( K / L ) ) + 1 ) ) )
3		simpl	|- ( ( ( |_ ` ( K / L ) ) <_ ( K / L ) /\ ( K / L ) < ( ( |_ ` ( K / L ) ) + 1 ) ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
4	1 2 3	3syl	|- ( ( K e. RR /\ L e. RR+ ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )