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Metamath Proof Explorer

Theorem fourierdlem8

Description: A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fourierdlem8.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
fourierdlem8.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
fourierdlem8.q ⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
fourierdlem8.i ⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ 𝑀 ) )
Assertion fourierdlem8 ( 𝜑 → ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )

Proof

Step Hyp Ref Expression
1 fourierdlem8.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2 fourierdlem8.b ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3 fourierdlem8.q ⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
4 fourierdlem8.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 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ( 𝐴 [,] 𝐵 ) )
8 4 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝐼 ∈ ( 0 ..^ 𝑀 ) )
9 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) )
10 5 6 7 8 9 fourierdlem1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
11 10 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
12 dfss3 ⊢ ( ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ↔ ∀ 𝑥 ∈ ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
13 11 12 sylibr ⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝐼 ) [,] ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )