Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem27.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
fourierdlem27.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
3 |
|
fourierdlem27.q |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) ) |
4 |
|
fourierdlem27.i |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ 𝑀 ) ) |
5 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝐴 ∈ ℝ* ) |
6 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝐵 ∈ ℝ* ) |
7 |
|
elioore |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → 𝑥 ∈ ℝ ) |
8 |
7
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 ∈ ℝ ) |
9 |
|
iccssxr |
⊢ ( 𝐴 [,] 𝐵 ) ⊆ ℝ* |
10 |
|
elfzofz |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝑀 ) → 𝐼 ∈ ( 0 ... 𝑀 ) ) |
11 |
4 10
|
syl |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ... 𝑀 ) ) |
12 |
3 11
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ∈ ( 𝐴 [,] 𝐵 ) ) |
13 |
9 12
|
sselid |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ) |
15 |
8
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 ∈ ℝ* ) |
16 |
|
iccgelb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑄 ‘ 𝐼 ) ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ ( 𝑄 ‘ 𝐼 ) ) |
17 |
1 2 12 16
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ≤ ( 𝑄 ‘ 𝐼 ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝐴 ≤ ( 𝑄 ‘ 𝐼 ) ) |
19 |
|
fzofzp1 |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝑀 ) → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
20 |
4 19
|
syl |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
21 |
3 20
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
22 |
9 21
|
sselid |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ) |
24 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
25 |
|
ioogtlb |
⊢ ( ( ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ( 𝑄 ‘ 𝐼 ) < 𝑥 ) |
26 |
14 23 24 25
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ( 𝑄 ‘ 𝐼 ) < 𝑥 ) |
27 |
5 14 15 18 26
|
xrlelttrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝐴 < 𝑥 ) |
28 |
|
iooltub |
⊢ ( ( ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
29 |
14 23 24 28
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
30 |
|
iccleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ 𝐵 ) |
31 |
1 2 21 30
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ 𝐵 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ 𝐵 ) |
33 |
15 23 6 29 32
|
xrltletrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 < 𝐵 ) |
34 |
5 6 8 27 33
|
eliood |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) |
35 |
34
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) |
36 |
|
dfss3 |
⊢ ( ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ⊆ ( 𝐴 (,) 𝐵 ) ↔ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) |
37 |
35 36
|
sylibr |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ⊆ ( 𝐴 (,) 𝐵 ) ) |