This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem chpwordi

Description: The second Chebyshev function is weakly increasing. (Contributed by Mario Carneiro, 9-Apr-2016)

Ref Expression
Assertion chpwordi ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ψ ‘ 𝐴 ) ≤ ( ψ ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 fzfid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( 1 ... ( ⌊ ‘ 𝐵 ) ) ∈ Fin )
2 elfznn ⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) → 𝑛 ∈ ℕ )
3 2 adantl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) ) → 𝑛 ∈ ℕ )
4 vmacl ⊢ ( 𝑛 ∈ ℕ → ( Λ ‘ 𝑛 ) ∈ ℝ )
5 3 4 syl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) ) → ( Λ ‘ 𝑛 ) ∈ ℝ )
6 vmage0 ⊢ ( 𝑛 ∈ ℕ → 0 ≤ ( Λ ‘ 𝑛 ) )
7 3 6 syl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) ) → 0 ≤ ( Λ ‘ 𝑛 ) )
8 flword2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ⌊ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ ( ⌊ ‘ 𝐴 ) ) )
9 fzss2 ⊢ ( ( ⌊ ‘ 𝐵 ) ∈ ( ℤ≥ ‘ ( ⌊ ‘ 𝐴 ) ) → ( 1 ... ( ⌊ ‘ 𝐴 ) ) ⊆ ( 1 ... ( ⌊ ‘ 𝐵 ) ) )
10 8 9 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( 1 ... ( ⌊ ‘ 𝐴 ) ) ⊆ ( 1 ... ( ⌊ ‘ 𝐵 ) ) )
11 1 5 7 10 fsumless ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) ≤ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) ( Λ ‘ 𝑛 ) )
12 chpval ⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) )
13 12 3ad2ant1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ψ ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) )
14 chpval ⊢ ( 𝐵 ∈ ℝ → ( ψ ‘ 𝐵 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) ( Λ ‘ 𝑛 ) )
15 14 3ad2ant2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ψ ‘ 𝐵 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐵 ) ) ( Λ ‘ 𝑛 ) )
16 11 13 15 3brtr4d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( ψ ‘ 𝐴 ) ≤ ( ψ ‘ 𝐵 ) )