Step |
Hyp |
Ref |
Expression |
1 |
|
sge0lessmpt.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
sge0lessmpt.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
3 |
|
sge0lessmpt.c |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 ) |
4 |
|
id |
⊢ ( 𝜑 → 𝜑 ) |
5 |
3
|
resmptd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) |
6 |
5
|
eqcomd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) |
7 |
4 6
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) |
8 |
7
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) = ( Σ^ ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) ) |
9 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
10 |
2 9
|
fmptd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
11 |
1 10
|
sge0less |
⊢ ( 𝜑 → ( Σ^ ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) ≤ ( Σ^ ‘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |
12 |
8 11
|
eqbrtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) ≤ ( Σ^ ‘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ) |