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Definition df-fl 9352
Description: Define the floor (greatest integer less than or equal to) function. See flval 9354 for its value, flqlelt 9358 for its basic property, and flqcl 9355 for its closure. For example,  ( |_ `  (
3  /  2 ) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 10741).

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 9350 . 2  class  |_
2 vx . . 3  setvar  x
3 cr 7042 . . 3  class  RR
4 vy . . . . . . 7  setvar  y
54cv 1284 . . . . . 6  class  y
62cv 1284 . . . . . 6  class  x
7 cle 7216 . . . . . 6  class  <_
85, 6, 7wbr 3793 . . . . 5  wff  y  <_  x
9 c1 7044 . . . . . . 7  class  1
10 caddc 7046 . . . . . . 7  class  +
115, 9, 10co 5543 . . . . . 6  class  ( y  +  1 )
12 clt 7215 . . . . . 6  class  <
136, 11, 12wbr 3793 . . . . 5  wff  x  < 
( y  +  1 )
148, 13wa 102 . . . 4  wff  ( y  <_  x  /\  x  <  ( y  +  1 ) )
15 cz 8432 . . . 4  class  ZZ
1614, 4, 15crio 5498 . . 3  class  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) )
172, 3, 16cmpt 3847 . 2  class  ( x  e.  RR  |->  ( iota_ y  e.  ZZ  ( y  <_  x  /\  x  <  ( y  +  1 ) ) ) )
181, 17wceq 1285 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  9354
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