Theorem prodfmul

Description: The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017)

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

Proof

Step	Hyp	Ref	Expression
1		prodfmul.1	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		prodfmul.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ )
3		prodfmul.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐺 ‘ 𝑘 ) ∈ ℂ )
4		prodfmul.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ 𝑘 ) ) )
5		mulcl	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
6	5	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
7		mulcom	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 · 𝑦 ) = ( 𝑦 · 𝑥 ) )
8	7	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 · 𝑦 ) = ( 𝑦 · 𝑥 ) )
9		mulass	⊢ ( ( 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 ) ) )