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Metamath Proof Explorer

Theorem gsummulg

Description: Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014) (Revised by Mario Carneiro, 7-Jan-2015) (Revised by AV, 6-Jun-2019)

		Ref	Expression
	Hypotheses	gsummulg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
		gsummulg.z	⊢ 0 = ( 0_g ‘ 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 )
		gsummulg.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
		gsummulg.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ₀ )
	Assertion	gsummulg	⊢ ( 𝜑 → ( 𝐺 Σ_g ( 𝑘 ∈ 𝐴 ↦ ( 𝑁 · 𝑋 ) ) ) = ( 𝑁 · ( 𝐺 Σ_g ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		gsummulg.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsummulg.z	⊢ 0 = ( 0_g ‘ 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		gsummulg.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
8		gsummulg.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ₀ )
9	8	nn0zd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
10	8	olcd	⊢ ( 𝜑 → ( 𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ₀ ) )
11	1 2 3 4 5 6 7 9 10	gsummulglem	⊢ ( 𝜑 → ( 𝐺 Σ_g ( 𝑘 ∈ 𝐴 ↦ ( 𝑁 · 𝑋 ) ) ) = ( 𝑁 · ( 𝐺 Σ_g ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )