Metamath Proof Explorer


Theorem gsumf1o

Description: Re-index a finite group sum using a bijection. (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 )
gsumcl.w ( 𝜑𝐹 finSupp 0 )
gsumf1o.h ( 𝜑𝐻 : 𝐶1-1-onto𝐴 )
Assertion gsumf1o ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝐹𝐻 ) ) )

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 gsumcl.w ( 𝜑𝐹 finSupp 0 )
7 gsumf1o.h ( 𝜑𝐻 : 𝐶1-1-onto𝐴 )
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 gsumzf1o ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝐹𝐻 ) ) )