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Definition df-fl 10073
Description: Define the floor (greatest integer less than or equal to) function. See flval 10075 for its value, flqlelt 10079 for its basic property, and flqcl 10076 for its closure. For example,  ( |_ `  (
3  /  2 ) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 13106).

Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision.

The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)

Assertion
Ref Expression
df-fl  |-  |_  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) ) )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-fl
StepHypRef Expression
1 cfl 10071 . 2  class  |_
2 vx . . 3  setvar  x
3 cr 7642 . . 3  class  RR
4 vy . . . . . . 7  setvar  y
54cv 1331 . . . . . 6  class  y
62cv 1331 . . . . . 6  class  x
7 cle 7824 . . . . . 6  class  <_
85, 6, 7wbr 3936 . . . . 5  wff  y  <_  x
9 c1 7644 . . . . . . 7  class  1
10 caddc 7646 . . . . . . 7  class  +
115, 9, 10co 5781 . . . . . 6  class  ( y  +  1 )
12 clt 7823 . . . . . 6  class  <
136, 11, 12wbr 3936 . . . . 5  wff  x  < 
( y  +  1 )
148, 13wa 103 . . . 4  wff  ( y  <_  x  /\  x  <  ( y  +  1 ) )
15 cz 9077 . . . 4  class  ZZ
1614, 4, 15crio 5736 . . 3  class  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) )
172, 3, 16cmpt 3996 . 2  class  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) ) )
181, 17wceq 1332 1  wff  |_  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  flval  10075
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