Metamath Proof Explorer
Description: Lemma for knoppndv . (Contributed by Asger C. Ipsen, 15-Jun-2021)
(Revised by Asger C. Ipsen, 5-Jul-2021)
|
|
Ref |
Expression |
|
Hypotheses |
knoppndvlem1.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
|
|
knoppndvlem1.j |
⊢ ( 𝜑 → 𝐽 ∈ ℤ ) |
|
|
knoppndvlem1.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
|
Assertion |
knoppndvlem1 |
⊢ ( 𝜑 → ( ( ( ( 2 · 𝑁 ) ↑ - 𝐽 ) / 2 ) · 𝑀 ) ∈ ℝ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
knoppndvlem1.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
knoppndvlem1.j |
⊢ ( 𝜑 → 𝐽 ∈ ℤ ) |
3 |
|
knoppndvlem1.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
4 |
|
2re |
⊢ 2 ∈ ℝ |
5 |
4
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ ) |
6 |
|
nnz |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
8 |
7
|
zred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
9 |
5 8
|
remulcld |
⊢ ( 𝜑 → ( 2 · 𝑁 ) ∈ ℝ ) |
10 |
5
|
recnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
11 |
8
|
recnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
12 |
|
2ne0 |
⊢ 2 ≠ 0 |
13 |
12
|
a1i |
⊢ ( 𝜑 → 2 ≠ 0 ) |
14 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
15 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
16 |
|
0lt1 |
⊢ 0 < 1 |
17 |
16
|
a1i |
⊢ ( 𝜑 → 0 < 1 ) |
18 |
|
nnge1 |
⊢ ( 𝑁 ∈ ℕ → 1 ≤ 𝑁 ) |
19 |
1 18
|
syl |
⊢ ( 𝜑 → 1 ≤ 𝑁 ) |
20 |
14 15 8 17 19
|
ltletrd |
⊢ ( 𝜑 → 0 < 𝑁 ) |
21 |
14 20
|
ltned |
⊢ ( 𝜑 → 0 ≠ 𝑁 ) |
22 |
21
|
necomd |
⊢ ( 𝜑 → 𝑁 ≠ 0 ) |
23 |
10 11 13 22
|
mulne0d |
⊢ ( 𝜑 → ( 2 · 𝑁 ) ≠ 0 ) |
24 |
2
|
znegcld |
⊢ ( 𝜑 → - 𝐽 ∈ ℤ ) |
25 |
9 23 24
|
reexpclzd |
⊢ ( 𝜑 → ( ( 2 · 𝑁 ) ↑ - 𝐽 ) ∈ ℝ ) |
26 |
25 5 13
|
redivcld |
⊢ ( 𝜑 → ( ( ( 2 · 𝑁 ) ↑ - 𝐽 ) / 2 ) ∈ ℝ ) |
27 |
3
|
zred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
28 |
26 27
|
remulcld |
⊢ ( 𝜑 → ( ( ( ( 2 · 𝑁 ) ↑ - 𝐽 ) / 2 ) · 𝑀 ) ∈ ℝ ) |