Description: Domain and range of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | ppif | ⊢ π : ℝ ⟶ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ppi | ⊢ π = ( 𝑥 ∈ ℝ ↦ ( ♯ ‘ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ) ) | |
2 | ppifi | ⊢ ( 𝑥 ∈ ℝ → ( ( 0 [,] 𝑥 ) ∩ ℙ ) ∈ Fin ) | |
3 | hashcl | ⊢ ( ( ( 0 [,] 𝑥 ) ∩ ℙ ) ∈ Fin → ( ♯ ‘ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ) ∈ ℕ0 ) | |
4 | 2 3 | syl | ⊢ ( 𝑥 ∈ ℝ → ( ♯ ‘ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ) ∈ ℕ0 ) |
5 | 1 4 | fmpti | ⊢ π : ℝ ⟶ ℕ0 |