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Metamath Proof Explorer

Theorem flid

Description: An integer is its own floor. (Contributed by NM, 15-Nov-2004)

Ref Expression
Assertion flid 
|- ( A e. ZZ -> ( |_ ` A ) = A )

Proof

Step Hyp Ref Expression
1 zre 
 |-  ( A e. ZZ -> A e. RR )
2 flle 
 |-  ( A e. RR -> ( |_ ` A ) <_ A )
3 1 2 syl 
 |-  ( A e. ZZ -> ( |_ ` A ) <_ A )
4 1 leidd 
 |-  ( A e. ZZ -> A <_ A )
5 flge 
 |-  ( ( A e. RR /\ A e. ZZ ) -> ( A <_ A <-> A <_ ( |_ ` A ) ) )
6 1 5 mpancom 
 |-  ( A e. ZZ -> ( A <_ A <-> A <_ ( |_ ` A ) ) )
7 4 6 mpbid 
 |-  ( A e. ZZ -> A <_ ( |_ ` A ) )
8 reflcl 
 |-  ( A e. RR -> ( |_ ` A ) e. RR )
9 1 8 syl 
 |-  ( A e. ZZ -> ( |_ ` A ) e. RR )
10 9 1 letri3d 
 |-  ( A e. ZZ -> ( ( |_ ` A ) = A <-> ( ( |_ ` A ) <_ A /\ A <_ ( |_ ` A ) ) ) )
11 3 7 10 mpbir2and 
 |-  ( A e. ZZ -> ( |_ ` A ) = A )