Theorem flwordi

Description: Ordering relation for the floor function. (Contributed by NM, 31-Dec-2005) (Proof shortened by Fan Zheng, 14-Jul-2016)

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

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	2	zred	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` A ) e. RR )
4		simp2	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> B e. RR )
5		flle	|- ( A e. RR -> ( |_ ` A ) <_ A )
6	1 5	syl	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` A ) <_ A )
7		simp3	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> A <_ B )
8	3 1 4 6 7	letrd	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` A ) <_ B )
9		flge	|- ( ( B e. RR /\ ( |_ ` A ) e. ZZ ) -> ( ( |_ ` A ) <_ B <-> ( |_ ` A ) <_ ( |_ ` B ) ) )
10	4 2 9	syl2anc	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( |_ ` A ) <_ B <-> ( |_ ` A ) <_ ( |_ ` B ) ) )
11	8 10	mpbid	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` A ) <_ ( |_ ` B ) )