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Metamath Proof Explorer

Theorem ico01fl0

Description: The floor of a real number in [ 0 , 1 ) is 0. Remark: may shorten the proof of modid or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019)

Ref Expression
Assertion ico01fl0 
|- ( A e. ( 0 [,) 1 ) -> ( |_ ` A ) = 0 )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 1xr 
 |-  1 e. RR*
3 icossre 
 |-  ( ( 0 e. RR /\ 1 e. RR* ) -> ( 0 [,) 1 ) C_ RR )
4 1 2 3 mp2an 
 |-  ( 0 [,) 1 ) C_ RR
5 4 sseli 
 |-  ( A e. ( 0 [,) 1 ) -> A e. RR )
6 0xr 
 |-  0 e. RR*
7 elico1 
 |-  ( ( 0 e. RR* /\ 1 e. RR* ) -> ( A e. ( 0 [,) 1 ) <-> ( A e. RR* /\ 0 <_ A /\ A < 1 ) ) )
8 6 2 7 mp2an 
 |-  ( A e. ( 0 [,) 1 ) <-> ( A e. RR* /\ 0 <_ A /\ A < 1 ) )
9 8 simp2bi 
 |-  ( A e. ( 0 [,) 1 ) -> 0 <_ A )
10 8 simp3bi 
 |-  ( A e. ( 0 [,) 1 ) -> A < 1 )
11 recn 
 |-  ( A e. RR -> A e. CC )
12 11 addlidd 
 |-  ( A e. RR -> ( 0 + A ) = A )
13 12 fveqeq2d 
 |-  ( A e. RR -> ( ( |_ ` ( 0 + A ) ) = 0 <-> ( |_ ` A ) = 0 ) )
14 0z 
 |-  0 e. ZZ
15 flbi2 
 |-  ( ( 0 e. ZZ /\ A e. RR ) -> ( ( |_ ` ( 0 + A ) ) = 0 <-> ( 0 <_ A /\ A < 1 ) ) )
16 14 15 mpan 
 |-  ( A e. RR -> ( ( |_ ` ( 0 + A ) ) = 0 <-> ( 0 <_ A /\ A < 1 ) ) )
17 13 16 bitr3d 
 |-  ( A e. RR -> ( ( |_ ` A ) = 0 <-> ( 0 <_ A /\ A < 1 ) ) )
18 17 biimpar 
 |-  ( ( A e. RR /\ ( 0 <_ A /\ A < 1 ) ) -> ( |_ ` A ) = 0 )
19 5 9 10 18 syl12anc 
 |-  ( A e. ( 0 [,) 1 ) -> ( |_ ` A ) = 0 )