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Definition df-fl 12983
 Description: Define the floor (greatest integer less than or equal to) function. See flval 12985 for its value, fllelt 12988 for its basic property, and flcl 12986 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28019). 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 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-fl
StepHypRef Expression
1 cfl 12981 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 10340 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1507 . . . . . 6 class 𝑦
62cv 1507 . . . . . 6 class 𝑥
7 cle 10481 . . . . . 6 class
85, 6, 7wbr 4934 . . . . 5 wff 𝑦𝑥
9 c1 10342 . . . . . . 7 class 1
10 caddc 10344 . . . . . . 7 class +
115, 9, 10co 6982 . . . . . 6 class (𝑦 + 1)
12 clt 10480 . . . . . 6 class <
136, 11, 12wbr 4934 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 387 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11799 . . . 4 class
1614, 4, 15crio 6942 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 5013 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1508 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
 Colors of variables: wff setvar class This definition is referenced by:  flval  12985
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