Metamath Proof Explorer

Theorem leceifl

Description: Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017)

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

Proof

Step Hyp Ref Expression
1 ltflcei 
 |-  ( ( B e. RR /\ A e. RR ) -> ( ( |_ ` B ) < A <-> B < -u ( |_ ` -u A ) ) )
2 1 ancoms 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( |_ ` B ) < A <-> B < -u ( |_ ` -u A ) ) )
3 2 notbid 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -. ( |_ ` B ) < A <-> -. B < -u ( |_ ` -u A ) ) )
4 reflcl 
 |-  ( B e. RR -> ( |_ ` B ) e. RR )
5 lenlt 
 |-  ( ( A e. RR /\ ( |_ ` B ) e. RR ) -> ( A <_ ( |_ ` B ) <-> -. ( |_ ` B ) < A ) )
6 4 5 sylan2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ ( |_ ` B ) <-> -. ( |_ ` B ) < A ) )
7 ceicl 
 |-  ( A e. RR -> -u ( |_ ` -u A ) e. ZZ )
8 7 zred 
 |-  ( A e. RR -> -u ( |_ ` -u A ) e. RR )
9 lenlt 
 |-  ( ( -u ( |_ ` -u A ) e. RR /\ B e. RR ) -> ( -u ( |_ ` -u A ) <_ B <-> -. B < -u ( |_ ` -u A ) ) )
10 8 9 sylan 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u ( |_ ` -u A ) <_ B <-> -. B < -u ( |_ ` -u A ) ) )
11 3 6 10 3bitr4rd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u ( |_ ` -u A ) <_ B <-> A <_ ( |_ ` B ) ) )