Theorem ppival

Description: Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014)

	Ref	Expression
Assertion	ppival	⊢ ( 𝐴 ∈ ℝ → ( π ‘ 𝐴 ) = ( ♯ ‘ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑥 = 𝐴 → ( 0 [,] 𝑥 ) = ( 0 [,] 𝐴 ) )
2	1	ineq1d	⊢ ( 𝑥 = 𝐴 → ( ( 0 [,] 𝑥 ) ∩ ℙ ) = ( ( 0 [,] 𝐴 ) ∩ ℙ ) )
3	2	fveq2d	⊢ ( 𝑥 = 𝐴 → ( ♯ ‘ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ) = ( ♯ ‘ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) )
4		df-ppi	⊢ π = ( 𝑥 ∈ ℝ ↦ ( ♯ ‘ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ) )
5		fvex	⊢ ( ♯ ‘ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∈ V
6	3 4 5	fvmpt	⊢ ( 𝐴 ∈ ℝ → ( π ‘ 𝐴 ) = ( ♯ ‘ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) )