Theorem gsummptfsres

Description: Extend a finitely supported group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 15-Feb-2026)

	Ref	Expression
Hypotheses	gsummptfsres.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	gsummptfsres.2	⊢ 0 = ( 0g ‘ 𝐺 )
	gsummptfsres.3	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
	gsummptfsres.4	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gsummptfsres.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝑆 ) ) → 𝑌 = 0 )
	gsummptfsres.6	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) finSupp 0 )
	gsummptfsres.7	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 )
	gsummptfsres.8	⊢ ( 𝜑 → 𝑆 ⊆ 𝐴 )
Assertion	gsummptfsres	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		gsummptfsres.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsummptfsres.2	⊢ 0 = ( 0g ‘ 𝐺 )
3		gsummptfsres.3	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4		gsummptfsres.4	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5		gsummptfsres.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝑆 ) ) → 𝑌 = 0 )
6		gsummptfsres.6	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) finSupp 0 )
7		gsummptfsres.7	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 )
8		gsummptfsres.8	⊢ ( 𝜑 → 𝑆 ⊆ 𝐴 )
9	7	fmpttd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) : 𝐴 ⟶ 𝐵 )
10	5 4	suppss2	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) supp 0 ) ⊆ 𝑆 )
11	1 2 3 4 9 10 6	gsumres	⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ↾ 𝑆 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ) )
12	8	resmptd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ↾ 𝑆 ) = ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) )
13	12	oveq2d	⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ↾ 𝑆 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) ) )
14	11 13	eqtr3d	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) ) )