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Definition df-fl 10288
Description: Define the floor (greatest integer less than or equal to) function. See flval 10290 for its value, flqlelt 10294 for its basic property, and flqcl 10291 for its closure. For example,  ( |_ `  (
3  /  2 ) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 14874).

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 10286 . 2  class  |_
2 vx . . 3  setvar  x
3 cr 7828 . . 3  class  RR
4 vy . . . . . . 7  setvar  y
54cv 1363 . . . . . 6  class  y
62cv 1363 . . . . . 6  class  x
7 cle 8011 . . . . . 6  class  <_
85, 6, 7wbr 4018 . . . . 5  wff  y  <_  x
9 c1 7830 . . . . . . 7  class  1
10 caddc 7832 . . . . . . 7  class  +
115, 9, 10co 5891 . . . . . 6  class  ( y  +  1 )
12 clt 8010 . . . . . 6  class  <
136, 11, 12wbr 4018 . . . . 5  wff  x  < 
( y  +  1 )
148, 13wa 104 . . . 4  wff  ( y  <_  x  /\  x  <  ( y  +  1 ) )
15 cz 9271 . . . 4  class  ZZ
1614, 4, 15crio 5846 . . 3  class  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) )
172, 3, 16cmpt 4079 . 2  class  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) ) )
181, 17wceq 1364 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  10290
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