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