Description: A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | sumsns | ⊢ ( ( 𝑀 ∈ 𝑉 ∧ ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ∈ ℂ ) → Σ 𝑘 ∈ { 𝑀 } 𝐴 = ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv | ⊢ Ⅎ 𝑛 𝐴 | |
2 | nfcsb1v | ⊢ Ⅎ 𝑘 ⦋ 𝑛 / 𝑘 ⦌ 𝐴 | |
3 | csbeq1a | ⊢ ( 𝑘 = 𝑛 → 𝐴 = ⦋ 𝑛 / 𝑘 ⦌ 𝐴 ) | |
4 | 1 2 3 | cbvsumi | ⊢ Σ 𝑘 ∈ { 𝑀 } 𝐴 = Σ 𝑛 ∈ { 𝑀 } ⦋ 𝑛 / 𝑘 ⦌ 𝐴 |
5 | csbeq1 | ⊢ ( 𝑛 = 𝑀 → ⦋ 𝑛 / 𝑘 ⦌ 𝐴 = ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ) | |
6 | 5 | sumsn | ⊢ ( ( 𝑀 ∈ 𝑉 ∧ ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ∈ ℂ ) → Σ 𝑛 ∈ { 𝑀 } ⦋ 𝑛 / 𝑘 ⦌ 𝐴 = ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ) |
7 | 4 6 | eqtrid | ⊢ ( ( 𝑀 ∈ 𝑉 ∧ ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ∈ ℂ ) → Σ 𝑘 ∈ { 𝑀 } 𝐴 = ⦋ 𝑀 / 𝑘 ⦌ 𝐴 ) |