Theorem ovolshft

Description: The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014) (Proof shortened by AV, 17-Sep-2020)

	Ref	Expression
Hypotheses	ovolshft.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
	ovolshft.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	ovolshft.3	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } )
Assertion	ovolshft	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = ( vol* ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ovolshft.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
2		ovolshft.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		ovolshft.3	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } )
4		eqid	⊢ { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } = { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) }
5	1 2 3 4	ovolshftlem2	⊢ ( 𝜑 → { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } ⊆ { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } )
6		ssrab2	⊢ { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } ⊆ ℝ
7	3 6	eqsstrdi	⊢ ( 𝜑 → 𝐵 ⊆ ℝ )
8	2	renegcld	⊢ ( 𝜑 → - 𝐶 ∈ ℝ )
9	1 2 3	shft2rab	⊢ ( 𝜑 → 𝐴 = { 𝑤 ∈ ℝ ∣ ( 𝑤 − - 𝐶 ) ∈ 𝐵 } )
10		eqid	⊢ { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ 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* , < ) ) }
11	7 8 9 10	ovolshftlem2	⊢ ( 𝜑 → { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } ⊆ { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } )
12	5 11	eqssd	⊢ ( 𝜑 → { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } = { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } )
13	12	infeq1d	⊢ ( 𝜑 → inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } , ℝ* , < ) = inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } , ℝ* , < ) )
14	10	ovolval	⊢ ( 𝐴 ⊆ ℝ → ( vol* ‘ 𝐴 ) = inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } , ℝ* , < ) )
15	1 14	syl	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = inf ( { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } , ℝ* , < ) )
16	4	ovolval	⊢ ( 𝐵 ⊆ ℝ → ( vol* ‘ 𝐵 ) = inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } , ℝ* , < ) )
17	7 16	syl	⊢ ( 𝜑 → ( vol* ‘ 𝐵 ) = inf ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑔 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑔 ) ∧ 𝑧 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑔 ) ) , ℝ* , < ) ) } , ℝ* , < ) )
18	13 15 17	3eqtr4d	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = ( vol* ‘ 𝐵 ) )