Step |
Hyp |
Ref |
Expression |
1 |
|
iccvonmbl.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
iccvonmbl.s |
⊢ 𝑆 = dom ( voln ‘ 𝑋 ) |
3 |
|
iccvonmbl.a |
⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ ) |
4 |
|
iccvonmbl.b |
⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ ) |
5 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐴 ‘ 𝑗 ) = ( 𝐴 ‘ 𝑖 ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐴 ‘ 𝑗 ) − ( 1 / 𝑛 ) ) = ( ( 𝐴 ‘ 𝑖 ) − ( 1 / 𝑛 ) ) ) |
7 |
6
|
cbvmptv |
⊢ ( 𝑗 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑗 ) − ( 1 / 𝑛 ) ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑖 ) − ( 1 / 𝑛 ) ) ) |
8 |
7
|
mpteq2i |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑗 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑗 ) − ( 1 / 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑖 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑖 ) − ( 1 / 𝑛 ) ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝐵 ‘ 𝑗 ) = ( 𝐵 ‘ 𝑖 ) ) |
10 |
9
|
oveq1d |
⊢ ( 𝑗 = 𝑖 → ( ( 𝐵 ‘ 𝑗 ) + ( 1 / 𝑛 ) ) = ( ( 𝐵 ‘ 𝑖 ) + ( 1 / 𝑛 ) ) ) |
11 |
10
|
cbvmptv |
⊢ ( 𝑗 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑗 ) + ( 1 / 𝑛 ) ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑖 ) + ( 1 / 𝑛 ) ) ) |
12 |
11
|
mpteq2i |
⊢ ( 𝑛 ∈ ℕ ↦ ( 𝑗 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑗 ) + ( 1 / 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝑖 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑖 ) + ( 1 / 𝑛 ) ) ) ) |
13 |
1 2 3 4 8 12
|
iccvonmbllem |
⊢ ( 𝜑 → X 𝑖 ∈ 𝑋 ( ( 𝐴 ‘ 𝑖 ) [,] ( 𝐵 ‘ 𝑖 ) ) ∈ 𝑆 ) |