Metamath Proof Explorer


Theorem sge0snmptf

Description: A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 21-Nov-2020)

Ref Expression
Hypotheses sge0snmptf.k 𝑘 𝜑
sge0snmptf.a ( 𝜑𝐴𝑉 )
sge0snmptf.c ( 𝜑𝐶 ∈ ( 0 [,] +∞ ) )
sge0snmptf.b ( 𝑘 = 𝐴𝐵 = 𝐶 )
Assertion sge0snmptf ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ) = 𝐶 )

Proof

Step Hyp Ref Expression
1 sge0snmptf.k 𝑘 𝜑
2 sge0snmptf.a ( 𝜑𝐴𝑉 )
3 sge0snmptf.c ( 𝜑𝐶 ∈ ( 0 [,] +∞ ) )
4 sge0snmptf.b ( 𝑘 = 𝐴𝐵 = 𝐶 )
5 elsni ( 𝑘 ∈ { 𝐴 } → 𝑘 = 𝐴 )
6 5 4 syl ( 𝑘 ∈ { 𝐴 } → 𝐵 = 𝐶 )
7 6 adantl ( ( 𝜑𝑘 ∈ { 𝐴 } ) → 𝐵 = 𝐶 )
8 3 adantr ( ( 𝜑𝑘 ∈ { 𝐴 } ) → 𝐶 ∈ ( 0 [,] +∞ ) )
9 7 8 eqeltrd ( ( 𝜑𝑘 ∈ { 𝐴 } ) → 𝐵 ∈ ( 0 [,] +∞ ) )
10 eqid ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 )
11 1 9 10 fmptdf ( 𝜑 → ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) : { 𝐴 } ⟶ ( 0 [,] +∞ ) )
12 2 11 sge0sn ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ) = ( ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ‘ 𝐴 ) )
13 snidg ( 𝐴𝑉𝐴 ∈ { 𝐴 } )
14 2 13 syl ( 𝜑𝐴 ∈ { 𝐴 } )
15 10 4 14 3 fvmptd3 ( 𝜑 → ( ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ‘ 𝐴 ) = 𝐶 )
16 12 15 eqtrd ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 } ↦ 𝐵 ) ) = 𝐶 )