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Metamath Proof Explorer

Theorem seradd

Description: The sum of two infinite series. (Contributed by NM, 17-Mar-2005) (Revised by Mario Carneiro, 26-May-2014)

Ref Expression
Hypotheses seradd.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
seradd.2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
seradd.3 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ )
seradd.4 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) + ( 𝐺 ‘ 𝑘 ) ) )
Assertion seradd ( 𝜑 → ( seq 𝑀 ( + , 𝐻 ) ‘ 𝑁 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 seradd.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2 seradd.2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
3 seradd.3 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ )
4 seradd.4 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) + ( 𝐺 ‘ 𝑘 ) ) )
5 addcl ⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + 𝑦 ) ∈ ℂ )
6 5 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 + 𝑦 ) ∈ ℂ )
7 addcom ⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
8 7 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
9 addass ⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
10 9 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
11 6 8 10 1 2 3 4 seqcaopr ⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐻 ) ‘ 𝑁 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) + ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) )