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Metamath Proof Explorer

Theorem flltnz

Description: The floor of a non-integer real is less than it. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

Step Hyp Ref Expression
1 reflcl 
 |-  ( A e. RR -> ( |_ ` A ) e. RR )
2 1 adantr 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> ( |_ ` A ) e. RR )
3 simpl 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> A e. RR )
4 fllelt 
 |-  ( A e. RR -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
5 4 adantr 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
6 5 simpld 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> ( |_ ` A ) <_ A )
7 simpr 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> -. A e. ZZ )
8 flidz 
 |-  ( A e. RR -> ( ( |_ ` A ) = A <-> A e. ZZ ) )
9 8 adantr 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> ( ( |_ ` A ) = A <-> A e. ZZ ) )
10 7 9 mtbird 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> -. ( |_ ` A ) = A )
11 10 neqned 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> ( |_ ` A ) =/= A )
12 11 necomd 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> A =/= ( |_ ` A ) )
13 2 3 6 12 leneltd 
 |-  ( ( A e. RR /\ -. A e. ZZ ) -> ( |_ ` A ) < A )