Definition df-ppi 27010

Description: Define the prime π function, which counts the number of primes less than or equal to 𝑥, see definition in [ApostolNT] p. 8. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
df-ppi	⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))

Detailed syntax breakdown of Definition df-ppi

Step	Hyp	Ref	Expression
1		cppi 27004	. 2 class π
2		vx	. . 3 setvar 𝑥
3		cr 11067	. . 3 class ℝ
4		cc0 11068	. . . . . 6 class 0
5	2	cv 1539	. . . . . 6 class 𝑥
6		cicc 13309	. . . . . 6 class [,]
7	4, 5, 6	co 7387	. . . . 5 class (0[,]𝑥)
8		cprime 16641	. . . . 5 class ℙ
9	7, 8	cin 3913	. . . 4 class ((0[,]𝑥) ∩ ℙ)
10		chash 14295	. . . 4 class ♯
11	9, 10	cfv 6511	. . 3 class (♯‘((0[,]𝑥) ∩ ℙ))
12	2, 3, 11	cmpt 5188	. 2 class (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
13	1, 12	wceq 1540	1 wff π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))

This definition is referenced by:  ppival  27037  ppif  27040