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 = ( 0g ‘ 𝐺 )
	gsumcl.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
	gsumcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gsumcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
	gsumcl.w	⊢ ( 𝜑 → 𝐹 finSupp 0 )
	gsumf1o.h	⊢ ( 𝜑 → 𝐻 : 𝐶 –1-1-onto→ 𝐴 )
Assertion	gsumf1o	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝐹 ∘ 𝐻 ) ) )

Proof

Step	Hyp	Ref	Expression
1		gsumcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsumcl.z	⊢ 0 = ( 0g ‘ 𝐺 )
3		gsumcl.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4		gsumcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5		gsumcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
6		gsumcl.w	⊢ ( 𝜑 → 𝐹 finSupp 0 )
7		gsumf1o.h	⊢ ( 𝜑 → 𝐻 : 𝐶 –1-1-onto→ 𝐴 )
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	gsumzf1o	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝐹 ∘ 𝐻 ) ) )