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Metamath Proof Explorer

Theorem flge

Description: The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004) (Proof shortened by Fan Zheng, 14-Jul-2016)

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

Proof

Step Hyp Ref Expression
1 flltp1 
 |-  ( A e. RR -> A < ( ( |_ ` A ) + 1 ) )
2 1 adantr 
 |-  ( ( A e. RR /\ B e. ZZ ) -> A < ( ( |_ ` A ) + 1 ) )
3 simpr 
 |-  ( ( A e. RR /\ B e. ZZ ) -> B e. ZZ )
4 3 zred 
 |-  ( ( A e. RR /\ B e. ZZ ) -> B e. RR )
5 simpl 
 |-  ( ( A e. RR /\ B e. ZZ ) -> A e. RR )
6 5 flcld 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( |_ ` A ) e. ZZ )
7 6 peano2zd 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( ( |_ ` A ) + 1 ) e. ZZ )
8 7 zred 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( ( |_ ` A ) + 1 ) e. RR )
9 lelttr 
 |-  ( ( B e. RR /\ A e. RR /\ ( ( |_ ` A ) + 1 ) e. RR ) -> ( ( B <_ A /\ A < ( ( |_ ` A ) + 1 ) ) -> B < ( ( |_ ` A ) + 1 ) ) )
10 4 5 8 9 syl3anc 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( ( B <_ A /\ A < ( ( |_ ` A ) + 1 ) ) -> B < ( ( |_ ` A ) + 1 ) ) )
11 2 10 mpan2d 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ A -> B < ( ( |_ ` A ) + 1 ) ) )
12 zleltp1 
 |-  ( ( B e. ZZ /\ ( |_ ` A ) e. ZZ ) -> ( B <_ ( |_ ` A ) <-> B < ( ( |_ ` A ) + 1 ) ) )
13 3 6 12 syl2anc 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ ( |_ ` A ) <-> B < ( ( |_ ` A ) + 1 ) ) )
14 11 13 sylibrd 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ A -> B <_ ( |_ ` A ) ) )
15 flle 
 |-  ( A e. RR -> ( |_ ` A ) <_ A )
16 15 adantr 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( |_ ` A ) <_ A )
17 6 zred 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( |_ ` A ) e. RR )
18 letr 
 |-  ( ( B e. RR /\ ( |_ ` A ) e. RR /\ A e. RR ) -> ( ( B <_ ( |_ ` A ) /\ ( |_ ` A ) <_ A ) -> B <_ A ) )
19 4 17 5 18 syl3anc 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( ( B <_ ( |_ ` A ) /\ ( |_ ` A ) <_ A ) -> B <_ A ) )
20 16 19 mpan2d 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ ( |_ ` A ) -> B <_ A ) )
21 14 20 impbid 
 |-  ( ( A e. RR /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) )