Theorem flword2

Description: Ordering relation for the floor function. (Contributed by Mario Carneiro, 7-Jun-2016)

	Ref	Expression
Assertion	flword2	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A e. RR )
2	1	flcld	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` A ) e. ZZ )
3		simp2	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> B e. RR )
4	3	flcld	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` B ) e. ZZ )
5		flwordi	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` A ) <_ ( |_ ` B ) )
6		eluz2	|- ( ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) <-> ( ( |_ ` A ) e. ZZ /\ ( |_ ` B ) e. ZZ /\ ( |_ ` A ) <_ ( |_ ` B ) ) )
7	2 4 5 6	syl3anbrc	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) )