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Metamath Proof Explorer

Theorem fsumsers

Description: Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013) (Revised by Mario Carneiro, 21-Apr-2014)

Ref Expression
Hypotheses fsumsers.1 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) )
fsumsers.2 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
fsumsers.3 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
fsumsers.4 ⊢ ( 𝜑 → 𝐴 ⊆ ( 𝑀 ... 𝑁 ) )
Assertion fsumsers ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 fsumsers.1 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) )
2 fsumsers.2 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
3 fsumsers.3 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
4 fsumsers.4 ⊢ ( 𝜑 → 𝐴 ⊆ ( 𝑀 ... 𝑁 ) )
5 eqid ⊢ ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝑀 )
6 eluzel2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
7 2 6 syl ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
8 fzssuz ⊢ ( 𝑀 ... 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 )
9 4 8 sstrdi ⊢ ( 𝜑 → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) )
10 5 7 9 1 3 zsum ⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘ seq 𝑀 ( + , 𝐹 ) ) )
11 fclim ⊢ ⇝ : dom ⇝ ⟶ ℂ
12 ffun ⊢ ( ⇝ : dom ⇝ ⟶ ℂ → Fun ⇝ )
13 11 12 ax-mp ⊢ Fun ⇝
14 1 2 3 4 fsumcvg2 ⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ⇝ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) )
15 funbrfv ⊢ ( Fun ⇝ → ( seq 𝑀 ( + , 𝐹 ) ⇝ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) → ( ⇝ ‘ seq 𝑀 ( + , 𝐹 ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) ) )
16 13 14 15 mpsyl ⊢ ( 𝜑 → ( ⇝ ‘ seq 𝑀 ( + , 𝐹 ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) )
17 10 16 eqtrd ⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) )