Theorem fllelt

Description: A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004) (Revised by Mario Carneiro, 2-Nov-2013)

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

Proof

Step	Hyp	Ref	Expression
1		flval	|- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
2	1	eqcomd	|- ( A e. RR -> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` A ) )
3		flcl	|- ( A e. RR -> ( |_ ` A ) e. ZZ )
4		rebtwnz	|- ( A e. RR -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
5		breq1	|- ( x = ( |_ ` A ) -> ( x <_ A <-> ( |_ ` A ) <_ A ) )
6		oveq1	|- ( x = ( |_ ` A ) -> ( x + 1 ) = ( ( |_ ` A ) + 1 ) )
7	6	breq2d	|- ( x = ( |_ ` A ) -> ( A < ( x + 1 ) <-> A < ( ( |_ ` A ) + 1 ) ) )
8	5 7	anbi12d	|- ( x = ( |_ ` A ) -> ( ( x <_ A /\ A < ( x + 1 ) ) <-> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) ) )
9	8	riota2	|- ( ( ( |_ ` A ) e. ZZ /\ E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) -> ( ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` A ) ) )
10	3 4 9	syl2anc	|- ( A e. RR -> ( ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` A ) ) )
11	2 10	mpbird	|- ( A e. RR -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )