Theorem hgt750lemc

Description: An upper bound to the summatory function of the von Mangoldt function. (Contributed by Thierry Arnoux, 29-Dec-2021)

	Ref	Expression
Hypothesis	hgt750lemc.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
Assertion	hgt750lemc	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑁 ) ( Λ ‘ 𝑗 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		hgt750lemc.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2	1	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3		chpvalz	⊢ ( 𝑁 ∈ ℤ → ( ψ ‘ 𝑁 ) = Σ 𝑗 ∈ ( 1 ... 𝑁 ) ( Λ ‘ 𝑗 ) )
4	2 3	syl	⊢ ( 𝜑 → ( ψ ‘ 𝑁 ) = Σ 𝑗 ∈ ( 1 ... 𝑁 ) ( Λ ‘ 𝑗 ) )
5		fveq2	⊢ ( 𝑥 = 𝑁 → ( ψ ‘ 𝑥 ) = ( ψ ‘ 𝑁 ) )
6		oveq2	⊢ ( 𝑥 = 𝑁 → ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑥 ) = ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑁 ) )
7	5 6	breq12d	⊢ ( 𝑥 = 𝑁 → ( ( ψ ‘ 𝑥 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑥 ) ↔ ( ψ ‘ 𝑁 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑁 ) ) )
8		ax-ros335	⊢ ∀ 𝑥 ∈ ℝ+ ( ψ ‘ 𝑥 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑥 )
9	8	a1i	⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ( ψ ‘ 𝑥 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑥 ) )
10	1	nnrpd	⊢ ( 𝜑 → 𝑁 ∈ ℝ+ )
11	7 9 10	rspcdva	⊢ ( 𝜑 → ( ψ ‘ 𝑁 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑁 ) )
12	4 11	eqbrtrrd	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑁 ) ( Λ ‘ 𝑗 ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) · 𝑁 ) )