Theorem fllep1

Description: A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016)

	Ref	Expression
Assertion	fllep1	|- ( A e. RR -> A <_ ( ( |_ ` A ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		flltp1	|- ( A e. RR -> A < ( ( |_ ` A ) + 1 ) )
2		reflcl	|- ( A e. RR -> ( |_ ` A ) e. RR )
3		peano2re	|- ( ( |_ ` A ) e. RR -> ( ( |_ ` A ) + 1 ) e. RR )
4	2 3	syl	|- ( A e. RR -> ( ( |_ ` A ) + 1 ) e. RR )
5		ltle	|- ( ( A e. RR /\ ( ( |_ ` A ) + 1 ) e. RR ) -> ( A < ( ( |_ ` A ) + 1 ) -> A <_ ( ( |_ ` A ) + 1 ) ) )
6	4 5	mpdan	|- ( A e. RR -> ( A < ( ( |_ ` A ) + 1 ) -> A <_ ( ( |_ ` A ) + 1 ) ) )
7	1 6	mpd	|- ( A e. RR -> A <_ ( ( |_ ` A ) + 1 ) )