Theorem sersub

Description: The difference of two infinite series. (Contributed by NM, 17-Mar-2005) (Revised by Mario Carneiro, 27-May-2014)

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

Proof

Step	Hyp	Ref	Expression
1		sersub.1	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		sersub.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
3		sersub.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ )
4		sersub.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		subcl	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 − 𝑦 ) ∈ ℂ )
8	7	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 − 𝑦 ) ∈ ℂ )
9		addsub4	⊢ ( ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( 𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ ) ) → ( ( 𝑥 + 𝑦 ) − ( 𝑧 + 𝑤 ) ) = ( ( 𝑥 − 𝑧 ) + ( 𝑦 − 𝑤 ) ) )
10	9	eqcomd	⊢ ( ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( 𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ ) ) → ( ( 𝑥 − 𝑧 ) + ( 𝑦 − 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) − ( 𝑧 + 𝑤 ) ) )
11	10	adantl	⊢ ( ( 𝜑 ∧ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( 𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ ) ) ) → ( ( 𝑥 − 𝑧 ) + ( 𝑦 − 𝑤 ) ) = ( ( 𝑥 + 𝑦 ) − ( 𝑧 + 𝑤 ) ) )
12	6 8 11 1 2 3 4	seqcaopr2	⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐻 ) ‘ 𝑁 ) = ( ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑁 ) − ( seq 𝑀 ( + , 𝐺 ) ‘ 𝑁 ) ) )