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 ) )