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Metamath Proof Explorer

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 ) )