Theorem ovolval

Description: The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014) (Revised by AV, 17-Sep-2020)

	Ref	Expression
Hypothesis	ovolval.1	⊢ 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) }
Assertion	ovolval	⊢ ( 𝐴 ⊆ ℝ → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) )

Proof

Step	Hyp	Ref	Expression
1		ovolval.1	⊢ 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) }
2		reex	⊢ ℝ ∈ V
3	2	elpw2	⊢ ( 𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ )
4		cleq1lem	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) ↔ ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) ) )
5	4	rexbidv	⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝑥 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) ↔ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) ) )
6	5	rabbidv	⊢ ( 𝑥 = 𝐴 → { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝑥 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } )
7	6 1	eqtr4di	⊢ ( 𝑥 = 𝐴 → { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝑥 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } = 𝑀 )
8	7	infeq1d	⊢ ( 𝑥 = 𝐴 → inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝑥 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } , ℝ* , < ) = inf ( 𝑀 , ℝ* , < ) )
9		df-ovol	⊢ vol* = ( 𝑥 ∈ 𝒫 ℝ ↦ inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝑥 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } , ℝ* , < ) )
10		xrltso	⊢ < Or ℝ*
11	10	infex	⊢ inf ( 𝑀 , ℝ* , < ) ∈ V
12	8 9 11	fvmpt	⊢ ( 𝐴 ∈ 𝒫 ℝ → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) )
13	3 12	sylbir	⊢ ( 𝐴 ⊆ ℝ → ( vol* ‘ 𝐴 ) = inf ( 𝑀 , ℝ* , < ) )