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 = ( 0g𝐺 )
gsumcl.g ( 𝜑𝐺 ∈ CMnd )
gsumcl.a ( 𝜑𝐴𝑉 )
gsumcl.f ( 𝜑𝐹 : 𝐴𝐵 )
gsumres.s ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝑊 )
gsumres.w ( 𝜑𝐹 finSupp 0 )
Assertion gsumres ( 𝜑 → ( 𝐺 Σg ( 𝐹𝑊 ) ) = ( 𝐺 Σg 𝐹 ) )

Proof

Step Hyp Ref Expression
1 gsumcl.b 𝐵 = ( Base ‘ 𝐺 )
2 gsumcl.z 0 = ( 0g𝐺 )
3 gsumcl.g ( 𝜑𝐺 ∈ CMnd )
4 gsumcl.a ( 𝜑𝐴𝑉 )
5 gsumcl.f ( 𝜑𝐹 : 𝐴𝐵 )
6 gsumres.s ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝑊 )
7 gsumres.w ( 𝜑𝐹 finSupp 0 )
8 eqid ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 )
9 cmnmnd ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
10 3 9 syl ( 𝜑𝐺 ∈ 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 Σg ( 𝐹𝑊 ) ) = ( 𝐺 Σg 𝐹 ) )