Metamath Proof Explorer


Theorem ovnsupge0

Description: The set used in the definition of the Lebesgue outer measure is a subset of the nonnegative extended reals. This is a substep for (a)(i) of the proof of Proposition 115D (a) of Fremlin1 p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Hypotheses ovnsupge0.1 ( 𝜑𝑋 ∈ Fin )
ovnsupge0.2 ( 𝜑𝐴 ⊆ ( ℝ ↑m 𝑋 ) )
ovnsupge0.3 𝑀 = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) }
Assertion ovnsupge0 ( 𝜑𝑀 ⊆ ( 0 [,] +∞ ) )

Proof

Step Hyp Ref Expression
1 ovnsupge0.1 ( 𝜑𝑋 ∈ Fin )
2 ovnsupge0.2 ( 𝜑𝐴 ⊆ ( ℝ ↑m 𝑋 ) )
3 ovnsupge0.3 𝑀 = { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) }
4 simp3 ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) )
5 nnex ℕ ∈ V
6 5 a1i ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) → ℕ ∈ V )
7 icossicc ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ )
8 nfv 𝑘 ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ )
9 1 ad2antrr ( ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → 𝑋 ∈ Fin )
10 elmapi ( 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → 𝑖 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) )
11 10 ad2antlr ( ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → 𝑖 : ℕ ⟶ ( ( ℝ × ℝ ) ↑m 𝑋 ) )
12 simpr ( ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ )
13 8 9 11 12 ovnprodcl ( ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ∈ ( 0 [,) +∞ ) )
14 7 13 sselid ( ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) ∧ 𝑗 ∈ ℕ ) → ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ∈ ( 0 [,] +∞ ) )
15 eqid ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) )
16 14 15 fmptd ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) → ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) : ℕ ⟶ ( 0 [,] +∞ ) )
17 6 16 sge0cl ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ∈ ( 0 [,] +∞ ) )
18 17 3adant3 ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ∈ ( 0 [,] +∞ ) )
19 4 18 eqeltrd ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) )
20 19 3adant3l ( ( 𝜑𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ∧ ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) )
21 20 3exp ( 𝜑 → ( 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → ( ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) ) ) )
22 21 adantr ( ( 𝜑𝑧 ∈ ℝ* ) → ( 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) → ( ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) ) ) )
23 22 rexlimdv ( ( 𝜑𝑧 ∈ ℝ* ) → ( ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) ) )
24 23 ralrimiva ( 𝜑 → ∀ 𝑧 ∈ ℝ* ( ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) ) )
25 rabss ( { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ( 0 [,] +∞ ) ↔ ∀ 𝑧 ∈ ℝ* ( ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) → 𝑧 ∈ ( 0 [,] +∞ ) ) )
26 24 25 sylibr ( 𝜑 → { 𝑧 ∈ ℝ* ∣ ∃ 𝑖 ∈ ( ( ( ℝ × ℝ ) ↑m 𝑋 ) ↑m ℕ ) ( 𝐴 𝑗 ∈ ℕ X 𝑘𝑋 ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ∧ 𝑧 = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ∏ 𝑘𝑋 ( vol ‘ ( ( [,) ∘ ( 𝑖𝑗 ) ) ‘ 𝑘 ) ) ) ) ) } ⊆ ( 0 [,] +∞ ) )
27 3 26 eqsstrid ( 𝜑𝑀 ⊆ ( 0 [,] +∞ ) )