This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fllt

Description: The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005)

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

Proof

Step Hyp Ref Expression
1 flge 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) )
2 zre 
 |-  ( B e. ZZ -> B e. RR )
3 lenlt 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B <_ A <-> -. A < B ) )
4 2 3 sylan 
 |-  ( ( B e. ZZ /\ A e. RR ) -> ( B <_ A <-> -. A < B ) )
5 4 ancoms 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ A <-> -. A < B ) )
6 reflcl 
 |-  ( A e. RR -> ( |_ ` A ) e. RR )
7 lenlt 
 |-  ( ( B e. RR /\ ( |_ ` A ) e. RR ) -> ( B <_ ( |_ ` A ) <-> -. ( |_ ` A ) < B ) )
8 2 6 7 syl2anr 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ ( |_ ` A ) <-> -. ( |_ ` A ) < B ) )
9 1 5 8 3bitr3d 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( -. A < B <-> -. ( |_ ` A ) < B ) )
10 9 con4bid 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) )