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Metamath Proof Explorer

Theorem gsumres

Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014) (Revised by Mario Carneiro, 24-Apr-2016) (Revised by AV, 3-Jun-2019)

		Ref	Expression
	Hypotheses	gsumcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
		gsumcl.z	⊢ 0 = ( 0_g ‘ 𝐺 )
		gsumcl.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
		gsumcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
		gsumcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
		gsumres.s	⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝑊 )
		gsumres.w	⊢ ( 𝜑 → 𝐹 finSupp 0 )
	Assertion	gsumres	⊢ ( 𝜑 → ( 𝐺 Σ_g ( 𝐹 ↾ 𝑊 ) ) = ( 𝐺 Σ_g 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		gsumcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsumcl.z	⊢ 0 = ( 0_g ‘ 𝐺 )
3		gsumcl.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4		gsumcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5		gsumcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
6		gsumres.s	⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝑊 )
7		gsumres.w	⊢ ( 𝜑 → 𝐹 finSupp 0 )
8		eqid	⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 )
9		cmnmnd	⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
10	3 9	syl	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
11	1 8 3 5	cntzcmnf	⊢ ( 𝜑 → ran 𝐹 ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran 𝐹 ) )
12	1 2 8 10 4 5 11 6 7	gsumzres	⊢ ( 𝜑 → ( 𝐺 Σ_g ( 𝐹 ↾ 𝑊 ) ) = ( 𝐺 Σ_g 𝐹 ) )