Metamath Proof Explorer

Theorem rddif

Description: The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 14-Sep-2015)

Ref Expression
Assertion rddif 
|- ( A e. RR -> ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) <_ ( 1 / 2 ) )

Proof

Step Hyp Ref Expression
1 halfcn 
 |-  ( 1 / 2 ) e. CC
2 1 2timesi 
 |-  ( 2 x. ( 1 / 2 ) ) = ( ( 1 / 2 ) + ( 1 / 2 ) )
3 2cn 
 |-  2 e. CC
4 2ne0 
 |-  2 =/= 0
5 3 4 recidi 
 |-  ( 2 x. ( 1 / 2 ) ) = 1
6 2 5 eqtr3i 
 |-  ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1
7 6 oveq2i 
 |-  ( ( A - ( 1 / 2 ) ) + ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( A - ( 1 / 2 ) ) + 1 )
8 recn 
 |-  ( A e. RR -> A e. CC )
9 1 a1i 
 |-  ( A e. RR -> ( 1 / 2 ) e. CC )
10 8 9 9 nppcan3d 
 |-  ( A e. RR -> ( ( A - ( 1 / 2 ) ) + ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( A + ( 1 / 2 ) ) )
11 7 10 eqtr3id 
 |-  ( A e. RR -> ( ( A - ( 1 / 2 ) ) + 1 ) = ( A + ( 1 / 2 ) ) )
12 halfre 
 |-  ( 1 / 2 ) e. RR
13 readdcl 
 |-  ( ( A e. RR /\ ( 1 / 2 ) e. RR ) -> ( A + ( 1 / 2 ) ) e. RR )
14 12 13 mpan2 
 |-  ( A e. RR -> ( A + ( 1 / 2 ) ) e. RR )
15 fllep1 
 |-  ( ( A + ( 1 / 2 ) ) e. RR -> ( A + ( 1 / 2 ) ) <_ ( ( |_ ` ( A + ( 1 / 2 ) ) ) + 1 ) )
16 14 15 syl 
 |-  ( A e. RR -> ( A + ( 1 / 2 ) ) <_ ( ( |_ ` ( A + ( 1 / 2 ) ) ) + 1 ) )
17 11 16 eqbrtrd 
 |-  ( A e. RR -> ( ( A - ( 1 / 2 ) ) + 1 ) <_ ( ( |_ ` ( A + ( 1 / 2 ) ) ) + 1 ) )
18 resubcl 
 |-  ( ( A e. RR /\ ( 1 / 2 ) e. RR ) -> ( A - ( 1 / 2 ) ) e. RR )
19 12 18 mpan2 
 |-  ( A e. RR -> ( A - ( 1 / 2 ) ) e. RR )
20 reflcl 
 |-  ( ( A + ( 1 / 2 ) ) e. RR -> ( |_ ` ( A + ( 1 / 2 ) ) ) e. RR )
21 14 20 syl 
 |-  ( A e. RR -> ( |_ ` ( A + ( 1 / 2 ) ) ) e. RR )
22 1red 
 |-  ( A e. RR -> 1 e. RR )
23 19 21 22 leadd1d 
 |-  ( A e. RR -> ( ( A - ( 1 / 2 ) ) <_ ( |_ ` ( A + ( 1 / 2 ) ) ) <-> ( ( A - ( 1 / 2 ) ) + 1 ) <_ ( ( |_ ` ( A + ( 1 / 2 ) ) ) + 1 ) ) )
24 17 23 mpbird 
 |-  ( A e. RR -> ( A - ( 1 / 2 ) ) <_ ( |_ ` ( A + ( 1 / 2 ) ) ) )
25 flle 
 |-  ( ( A + ( 1 / 2 ) ) e. RR -> ( |_ ` ( A + ( 1 / 2 ) ) ) <_ ( A + ( 1 / 2 ) ) )
26 14 25 syl 
 |-  ( A e. RR -> ( |_ ` ( A + ( 1 / 2 ) ) ) <_ ( A + ( 1 / 2 ) ) )
27 id 
 |-  ( A e. RR -> A e. RR )
28 12 a1i 
 |-  ( A e. RR -> ( 1 / 2 ) e. RR )
29 absdifle 
 |-  ( ( ( |_ ` ( A + ( 1 / 2 ) ) ) e. RR /\ A e. RR /\ ( 1 / 2 ) e. RR ) -> ( ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) <_ ( 1 / 2 ) <-> ( ( A - ( 1 / 2 ) ) <_ ( |_ ` ( A + ( 1 / 2 ) ) ) /\ ( |_ ` ( A + ( 1 / 2 ) ) ) <_ ( A + ( 1 / 2 ) ) ) ) )
30 21 27 28 29 syl3anc 
 |-  ( A e. RR -> ( ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) <_ ( 1 / 2 ) <-> ( ( A - ( 1 / 2 ) ) <_ ( |_ ` ( A + ( 1 / 2 ) ) ) /\ ( |_ ` ( A + ( 1 / 2 ) ) ) <_ ( A + ( 1 / 2 ) ) ) ) )
31 24 26 30 mpbir2and 
 |-  ( A e. RR -> ( abs ` ( ( |_ ` ( A + ( 1 / 2 ) ) ) - A ) ) <_ ( 1 / 2 ) )