Metamath Proof Explorer


Theorem knoppndvlem8

Description: Lemma for knoppndv . (Contributed by Asger C. Ipsen, 15-Jun-2021) (Revised by Asger C. Ipsen, 5-Jul-2021)

Ref Expression
Hypotheses knoppndvlem8.t 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) )
knoppndvlem8.f 𝐹 = ( 𝑦 ∈ ℝ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐶𝑛 ) · ( 𝑇 ‘ ( ( ( 2 · 𝑁 ) ↑ 𝑛 ) · 𝑦 ) ) ) ) )
knoppndvlem8.a 𝐴 = ( ( ( ( 2 · 𝑁 ) ↑ - 𝐽 ) / 2 ) · 𝑀 )
knoppndvlem8.c ( 𝜑𝐶 ∈ ( - 1 (,) 1 ) )
knoppndvlem8.j ( 𝜑𝐽 ∈ ℕ0 )
knoppndvlem8.m ( 𝜑𝑀 ∈ ℤ )
knoppndvlem8.n ( 𝜑𝑁 ∈ ℕ )
knoppndvlem8.1 ( 𝜑 → 2 ∥ 𝑀 )
Assertion knoppndvlem8 ( 𝜑 → ( ( 𝐹𝐴 ) ‘ 𝐽 ) = 0 )

Proof

Step Hyp Ref Expression
1 knoppndvlem8.t 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) )
2 knoppndvlem8.f 𝐹 = ( 𝑦 ∈ ℝ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐶𝑛 ) · ( 𝑇 ‘ ( ( ( 2 · 𝑁 ) ↑ 𝑛 ) · 𝑦 ) ) ) ) )
3 knoppndvlem8.a 𝐴 = ( ( ( ( 2 · 𝑁 ) ↑ - 𝐽 ) / 2 ) · 𝑀 )
4 knoppndvlem8.c ( 𝜑𝐶 ∈ ( - 1 (,) 1 ) )
5 knoppndvlem8.j ( 𝜑𝐽 ∈ ℕ0 )
6 knoppndvlem8.m ( 𝜑𝑀 ∈ ℤ )
7 knoppndvlem8.n ( 𝜑𝑁 ∈ ℕ )
8 knoppndvlem8.1 ( 𝜑 → 2 ∥ 𝑀 )
9 1 2 3 5 6 7 knoppndvlem7 ( 𝜑 → ( ( 𝐹𝐴 ) ‘ 𝐽 ) = ( ( 𝐶𝐽 ) · ( 𝑇 ‘ ( 𝑀 / 2 ) ) ) )
10 2z 2 ∈ ℤ
11 10 a1i ( 𝜑 → 2 ∈ ℤ )
12 2ne0 2 ≠ 0
13 12 a1i ( 𝜑 → 2 ≠ 0 )
14 11 13 6 3jca ( 𝜑 → ( 2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ ) )
15 dvdsval2 ( ( 2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ ) → ( 2 ∥ 𝑀 ↔ ( 𝑀 / 2 ) ∈ ℤ ) )
16 14 15 syl ( 𝜑 → ( 2 ∥ 𝑀 ↔ ( 𝑀 / 2 ) ∈ ℤ ) )
17 8 16 mpbid ( 𝜑 → ( 𝑀 / 2 ) ∈ ℤ )
18 1 17 dnizeq0 ( 𝜑 → ( 𝑇 ‘ ( 𝑀 / 2 ) ) = 0 )
19 18 oveq2d ( 𝜑 → ( ( 𝐶𝐽 ) · ( 𝑇 ‘ ( 𝑀 / 2 ) ) ) = ( ( 𝐶𝐽 ) · 0 ) )
20 4 knoppndvlem3 ( 𝜑 → ( 𝐶 ∈ ℝ ∧ ( abs ‘ 𝐶 ) < 1 ) )
21 20 simpld ( 𝜑𝐶 ∈ ℝ )
22 21 recnd ( 𝜑𝐶 ∈ ℂ )
23 22 5 expcld ( 𝜑 → ( 𝐶𝐽 ) ∈ ℂ )
24 23 mul01d ( 𝜑 → ( ( 𝐶𝐽 ) · 0 ) = 0 )
25 9 19 24 3eqtrd ( 𝜑 → ( ( 𝐹𝐴 ) ‘ 𝐽 ) = 0 )