Theorem ovnval2b

Description: Value of the Lebesgue outer measure of a subset A of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	ovnval2b.1	⊢ ( 𝜑 → 𝑋 ∈ Fin )
	ovnval2b.2	⊢ ( 𝜑 → 𝐴 ⊆ ( ℝ ↑m 𝑋 ) )
	ovnval2b.3	⊢ 𝐿 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } )
Assertion	ovnval2b	⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = if ( 𝑋 = ∅ , 0 , inf ( ( 𝐿 ‘ 𝐴 ) , ℝ* , < ) ) )

Proof

Step	Hyp	Ref	Expression
1		ovnval2b.1	⊢ ( 𝜑 → 𝑋 ∈ Fin )
2		ovnval2b.2	⊢ ( 𝜑 → 𝐴 ⊆ ( ℝ ↑m 𝑋 ) )
3		ovnval2b.3	⊢ 𝐿 = ( 𝑎 ∈ 𝒫 ( ℝ ↑m 𝑋 ) ↦ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } )
4		eqid	⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) }
5	1 2 4	ovnval2	⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = if ( 𝑋 = ∅ , 0 , inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) )
6		biidd	⊢ ( 𝜑 → ( 𝑋 = ∅ ↔ 𝑋 = ∅ ) )
7		cleq1lem	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) )
8	7	rexbidv	⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ↔ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) )
9	8	rabbidv	⊢ ( 𝑎 = 𝐴 → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } )
10		ovexd	⊢ ( 𝜑 → ( ℝ ↑m 𝑋 ) ∈ V )
11	10 2	ssexd	⊢ ( 𝜑 → 𝐴 ∈ V )
12	11 2	elpwd	⊢ ( 𝜑 → 𝐴 ∈ 𝒫 ( ℝ ↑m 𝑋 ) )
13		xrex	⊢ ℝ* ∈ V
14	13	rabex	⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ∈ V
15	14	a1i	⊢ ( 𝜑 → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ∈ V )
16	3 9 12 15	fvmptd3	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐴 ) = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } )
17	16	eqcomd	⊢ ( 𝜑 → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } = ( 𝐿 ‘ 𝐴 ) )
18	17	infeq1d	⊢ ( 𝜑 → inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) = inf ( ( 𝐿 ‘ 𝐴 ) , ℝ* , < ) )
19	6 18	ifbieq2d	⊢ ( 𝜑 → if ( 𝑋 = ∅ , 0 , inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X 𝑘 ∈ 𝑋 ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } , ℝ* , < ) ) = if ( 𝑋 = ∅ , 0 , inf ( ( 𝐿 ‘ 𝐴 ) , ℝ* , < ) ) )
20	5 19	eqtrd	⊢ ( 𝜑 → ( ( voln* ‘ 𝑋 ) ‘ 𝐴 ) = if ( 𝑋 = ∅ , 0 , inf ( ( 𝐿 ‘ 𝐴 ) , ℝ* , < ) ) )