Metamath Proof Explorer


Theorem iccvonmbl

Description: Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of Fremlin1 p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses iccvonmbl.x ( 𝜑𝑋 ∈ Fin )
iccvonmbl.s 𝑆 = dom ( voln ‘ 𝑋 )
iccvonmbl.a ( 𝜑𝐴 : 𝑋 ⟶ ℝ )
iccvonmbl.b ( 𝜑𝐵 : 𝑋 ⟶ ℝ )
Assertion iccvonmbl ( 𝜑X 𝑖𝑋 ( ( 𝐴𝑖 ) [,] ( 𝐵𝑖 ) ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 iccvonmbl.x ( 𝜑𝑋 ∈ Fin )
2 iccvonmbl.s 𝑆 = dom ( voln ‘ 𝑋 )
3 iccvonmbl.a ( 𝜑𝐴 : 𝑋 ⟶ ℝ )
4 iccvonmbl.b ( 𝜑𝐵 : 𝑋 ⟶ ℝ )
5 fveq2 ( 𝑗 = 𝑖 → ( 𝐴𝑗 ) = ( 𝐴𝑖 ) )
6 5 oveq1d ( 𝑗 = 𝑖 → ( ( 𝐴𝑗 ) − ( 1 / 𝑛 ) ) = ( ( 𝐴𝑖 ) − ( 1 / 𝑛 ) ) )
7 6 cbvmptv ( 𝑗𝑋 ↦ ( ( 𝐴𝑗 ) − ( 1 / 𝑛 ) ) ) = ( 𝑖𝑋 ↦ ( ( 𝐴𝑖 ) − ( 1 / 𝑛 ) ) )
8 7 mpteq2i ( 𝑛 ∈ ℕ ↦ ( 𝑗𝑋 ↦ ( ( 𝐴𝑗 ) − ( 1 / 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑖𝑋 ↦ ( ( 𝐴𝑖 ) − ( 1 / 𝑛 ) ) ) )
9 fveq2 ( 𝑗 = 𝑖 → ( 𝐵𝑗 ) = ( 𝐵𝑖 ) )
10 9 oveq1d ( 𝑗 = 𝑖 → ( ( 𝐵𝑗 ) + ( 1 / 𝑛 ) ) = ( ( 𝐵𝑖 ) + ( 1 / 𝑛 ) ) )
11 10 cbvmptv ( 𝑗𝑋 ↦ ( ( 𝐵𝑗 ) + ( 1 / 𝑛 ) ) ) = ( 𝑖𝑋 ↦ ( ( 𝐵𝑖 ) + ( 1 / 𝑛 ) ) )
12 11 mpteq2i ( 𝑛 ∈ ℕ ↦ ( 𝑗𝑋 ↦ ( ( 𝐵𝑗 ) + ( 1 / 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑖𝑋 ↦ ( ( 𝐵𝑖 ) + ( 1 / 𝑛 ) ) ) )
13 1 2 3 4 8 12 iccvonmbllem ( 𝜑X 𝑖𝑋 ( ( 𝐴𝑖 ) [,] ( 𝐵𝑖 ) ) ∈ 𝑆 )