Step |
Hyp |
Ref |
Expression |
1 |
|
sge0snmpt.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
sge0snmpt.c |
⊢ ( 𝜑 → 𝐶 ∈ ( 0 [,] +∞ ) ) |
3 |
|
sge0snmpt.b |
⊢ ( 𝑘 = 𝐴 → 𝐵 = 𝐶 ) |
4 |
|
elsni |
⊢ ( 𝑘 ∈ { 𝐴 } → 𝑘 = 𝐴 ) |
5 |
4 3
|
syl |
⊢ ( 𝑘 ∈ { 𝐴 } → 𝐵 = 𝐶 ) |
6 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → 𝐵 = 𝐶 ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
8 |
6 7
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝐴 } ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
9 |
|
eqid |
⊢ ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) |
10 |
8 9
|
fmptd |
⊢ ( 𝜑 → ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) : { 𝐴 } ⟶ ( 0 [,] +∞ ) ) |
11 |
1 10
|
sge0sn |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ) = ( ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ‘ 𝐴 ) ) |
12 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ) |
13 |
3
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐴 ) → 𝐵 = 𝐶 ) |
14 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 } ) |
15 |
1 14
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ { 𝐴 } ) |
16 |
12 13 15 2
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ‘ 𝐴 ) = 𝐶 ) |
17 |
11 16
|
eqtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ) = 𝐶 ) |