Theorem gsumreidx

Description: Re-index a finite group sum using a bijection. Corresponds to the first equation in Lang p. 5 with M = 1 . (Contributed by AV, 26-Dec-2023)

	Ref	Expression
Hypotheses	gsumreidx.b	$$⊢B={\mathrm{Base}}_G$$
	gsumreidx.z	$$⊢\underset{\dot{}}{0}=0_G$$
	gsumreidx.g	$$⊢\phi \to G\in \mathrm{CMnd}$$
	gsumreidx.f	$$⊢\phi \to F:\left(M\dots N\right)⟶B$$
	gsumreidx.h	$$⊢\phi \to H:\left(M\dots N\right)\underset{1-1\; onto}{⟶}\left(M\dots N\right)$$
Assertion	gsumreidx	$$⊢\phi \to {\sum }_{G}F={\sum }_G\left(F\circ H\right)$$

Proof

Step	Hyp	Ref	Expression
1		gsumreidx.b	$$⊢B={\mathrm{Base}}_G$$
2		gsumreidx.z	$$⊢\underset{\dot{}}{0}=0_G$$
3		gsumreidx.g	$$⊢\phi \to G\in \mathrm{CMnd}$$
4		gsumreidx.f	$$⊢\phi \to F:\left(M\dots N\right)⟶B$$
5		gsumreidx.h	$$⊢\phi \to H:\left(M\dots N\right)\underset{1-1\; onto}{⟶}\left(M\dots N\right)$$
6		ovexd	$$⊢\phi \to \left(M\dots N\right)\in \mathrm{V}$$
7		fzfid	$$⊢\phi \to \left(M\dots N\right)\in \mathrm{Fin}$$
8	2	fvexi	$$⊢\underset{\dot{}}{0}\in \mathrm{V}$$
9	8	a1i	$$⊢\phi \to \underset{\dot{}}{0}\in \mathrm{V}$$
10	4 7 9	fdmfifsupp	$$⊢\phi \to {finSupp}_{\underset{\dot{}}{0}}\left(F\right)$$
11	1 2 3 6 4 10 5	gsumf1o	$$⊢\phi \to {\sum }_{G}F={\sum }_G\left(F\circ H\right)$$