Theorem gsummulgz

Description: Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015) (Revised by AV, 6-Jun-2019)

	Ref	Expression
Hypotheses	gsummulg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	gsummulg.z	⊢ 0 = ( 0g ‘ 𝐺 )
	gsummulg.t	⊢ · = ( .g ‘ 𝐺 )
	gsummulg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gsummulg.f	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
	gsummulg.w	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
	gsummulgz.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
	gsummulgz.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion	gsummulgz	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑁 · 𝑋 ) ) ) = ( 𝑁 · ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		gsummulg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsummulg.z	⊢ 0 = ( 0g ‘ 𝐺 )
3		gsummulg.t	⊢ · = ( .g ‘ 𝐺 )
4		gsummulg.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5		gsummulg.f	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
6		gsummulg.w	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
7		gsummulgz.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
8		gsummulgz.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
9		ablcmn	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ CMnd )
10	7 9	syl	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
11	7	orcd	⊢ ( 𝜑 → ( 𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ0 ) )
12	1 2 3 4 5 6 10 8 11	gsummulglem	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑁 · 𝑋 ) ) ) = ( 𝑁 · ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )