Theorem ovolsca

Description: The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015)

	Ref	Expression
Hypotheses	ovolsca.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
	ovolsca.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	ovolsca.3	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∈ ℝ ∣ ( 𝐶 · 𝑥 ) ∈ 𝐴 } )
	ovolsca.4	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) ∈ ℝ )
Assertion	ovolsca	⊢ ( 𝜑 → ( vol* ‘ 𝐵 ) = ( ( vol* ‘ 𝐴 ) / 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ovolsca.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
2		ovolsca.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
3		ovolsca.3	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∈ ℝ ∣ ( 𝐶 · 𝑥 ) ∈ 𝐴 } )
4		ovolsca.4	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) ∈ ℝ )
5	1 2 3 4	ovolscalem2	⊢ ( 𝜑 → ( vol* ‘ 𝐵 ) ≤ ( ( vol* ‘ 𝐴 ) / 𝐶 ) )
6	4	recnd	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) ∈ ℂ )
7	2	rpcnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
8	2	rpne0d	⊢ ( 𝜑 → 𝐶 ≠ 0 )
9	6 7 8	divrecd	⊢ ( 𝜑 → ( ( vol* ‘ 𝐴 ) / 𝐶 ) = ( ( vol* ‘ 𝐴 ) · ( 1 / 𝐶 ) ) )
10		ssrab2	⊢ { 𝑥 ∈ ℝ ∣ ( 𝐶 · 𝑥 ) ∈ 𝐴 } ⊆ ℝ
11	3 10	eqsstrdi	⊢ ( 𝜑 → 𝐵 ⊆ ℝ )
12	2	rpreccld	⊢ ( 𝜑 → ( 1 / 𝐶 ) ∈ ℝ+ )
13	1 2 3	sca2rab	⊢ ( 𝜑 → 𝐴 = { 𝑦 ∈ ℝ ∣ ( ( 1 / 𝐶 ) · 𝑦 ) ∈ 𝐵 } )
14	4 2	rerpdivcld	⊢ ( 𝜑 → ( ( vol* ‘ 𝐴 ) / 𝐶 ) ∈ ℝ )
15		ovollecl	⊢ ( ( 𝐵 ⊆ ℝ ∧ ( ( vol* ‘ 𝐴 ) / 𝐶 ) ∈ ℝ ∧ ( vol* ‘ 𝐵 ) ≤ ( ( vol* ‘ 𝐴 ) / 𝐶 ) ) → ( vol* ‘ 𝐵 ) ∈ ℝ )
16	11 14 5 15	syl3anc	⊢ ( 𝜑 → ( vol* ‘ 𝐵 ) ∈ ℝ )
17	11 12 13 16	ovolscalem2	⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) ≤ ( ( vol* ‘ 𝐵 ) / ( 1 / 𝐶 ) ) )
18	4 16 12	lemuldivd	⊢ ( 𝜑 → ( ( ( vol* ‘ 𝐴 ) · ( 1 / 𝐶 ) ) ≤ ( vol* ‘ 𝐵 ) ↔ ( vol* ‘ 𝐴 ) ≤ ( ( vol* ‘ 𝐵 ) / ( 1 / 𝐶 ) ) ) )
19	17 18	mpbird	⊢ ( 𝜑 → ( ( vol* ‘ 𝐴 ) · ( 1 / 𝐶 ) ) ≤ ( vol* ‘ 𝐵 ) )
20	9 19	eqbrtrd	⊢ ( 𝜑 → ( ( vol* ‘ 𝐴 ) / 𝐶 ) ≤ ( vol* ‘ 𝐵 ) )
21	16 14	letri3d	⊢ ( 𝜑 → ( ( vol* ‘ 𝐵 ) = ( ( vol* ‘ 𝐴 ) / 𝐶 ) ↔ ( ( vol* ‘ 𝐵 ) ≤ ( ( vol* ‘ 𝐴 ) / 𝐶 ) ∧ ( ( vol* ‘ 𝐴 ) / 𝐶 ) ≤ ( vol* ‘ 𝐵 ) ) ) )
22	5 20 21	mpbir2and	⊢ ( 𝜑 → ( vol* ‘ 𝐵 ) = ( ( vol* ‘ 𝐴 ) / 𝐶 ) )