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Metamath Proof Explorer

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 ) )