Theorem flleceil

Description: The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018)

	Ref	Expression
Assertion	flleceil	|- ( A e. RR -> ( |_ ` A ) <_ ( |^ ` A ) )

Proof

Step	Hyp	Ref	Expression
1		reflcl	|- ( A e. RR -> ( |_ ` A ) e. RR )
2		id	|- ( A e. RR -> A e. RR )
3		ceilcl	|- ( A e. RR -> ( |^ ` A ) e. ZZ )
4	3	zred	|- ( A e. RR -> ( |^ ` A ) e. RR )
5		flle	|- ( A e. RR -> ( |_ ` A ) <_ A )
6		ceilge	|- ( A e. RR -> A <_ ( |^ ` A ) )
7	1 2 4 5 6	letrd	|- ( A e. RR -> ( |_ ` A ) <_ ( |^ ` A ) )