Theorem ppifl

Description: The prime-counting function ppi does not change off the integers. (Contributed by Mario Carneiro, 18-Sep-2014)

	Ref	Expression
Assertion	ppifl	⊢ ( 𝐴 ∈ ℝ → ( π ‘ ( ⌊ ‘ 𝐴 ) ) = ( π ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ppisval	⊢ ( 𝐴 ∈ ℝ → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) )
2	1	fveq2d	⊢ ( 𝐴 ∈ ℝ → ( ♯ ‘ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) = ( ♯ ‘ ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) )
3		ppival	⊢ ( 𝐴 ∈ ℝ → ( π ‘ 𝐴 ) = ( ♯ ‘ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) )
4		flcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℤ )
5		ppival2	⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ℤ → ( π ‘ ( ⌊ ‘ 𝐴 ) ) = ( ♯ ‘ ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) )
6	4 5	syl	⊢ ( 𝐴 ∈ ℝ → ( π ‘ ( ⌊ ‘ 𝐴 ) ) = ( ♯ ‘ ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) )
7	2 3 6	3eqtr4rd	⊢ ( 𝐴 ∈ ℝ → ( π ‘ ( ⌊ ‘ 𝐴 ) ) = ( π ‘ 𝐴 ) )