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Metamath Proof Explorer

Theorem fllelt

Description: A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004) (Revised by Mario Carneiro, 2-Nov-2013)

Ref Expression
Assertion fllelt 
|- ( A e. RR -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )

Proof

Step Hyp Ref Expression
1 flval 
 |-  ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
2 1 eqcomd 
 |-  ( A e. RR -> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` A ) )
3 flcl 
 |-  ( A e. RR -> ( |_ ` A ) e. ZZ )
4 rebtwnz 
 |-  ( A e. RR -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
5 breq1 
 |-  ( x = ( |_ ` A ) -> ( x <_ A <-> ( |_ ` A ) <_ A ) )
6 oveq1 
 |-  ( x = ( |_ ` A ) -> ( x + 1 ) = ( ( |_ ` A ) + 1 ) )
7 6 breq2d 
 |-  ( x = ( |_ ` A ) -> ( A < ( x + 1 ) <-> A < ( ( |_ ` A ) + 1 ) ) )
8 5 7 anbi12d 
 |-  ( x = ( |_ ` A ) -> ( ( x <_ A /\ A < ( x + 1 ) ) <-> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) ) )
9 8 riota2 
 |-  ( ( ( |_ ` A ) e. ZZ /\ E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) -> ( ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` A ) ) )
10 3 4 9 syl2anc 
 |-  ( A e. RR -> ( ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` A ) ) )
11 2 10 mpbird 
 |-  ( A e. RR -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )