Theorem plusffn

Description: The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015)

	Ref	Expression
Hypotheses	plusffn.1	$$⊢B={\mathrm{Base}}_G$$
	plusffn.2	$$⊢\underset{\dot{}}{⨣}={+}_{𝑓}\left(G\right)$$
Assertion	plusffn	$$⊢\underset{\dot{}}{⨣}Fn\left(B\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		plusffn.1	$$⊢B={\mathrm{Base}}_G$$
2		plusffn.2	$$⊢\underset{\dot{}}{⨣}={+}_{𝑓}\left(G\right)$$
3		eqid	$$⊢{+}_G={+}_G$$
4	1 3 2	plusffval	$$⊢\underset{\dot{}}{⨣}=\left(x\in B,y\in B⟼x{+}_{G}y\right)$$
5		ovex	$$⊢x{+}_{G}y\in \mathrm{V}$$
6	4 5	fnmpoi	$$⊢\underset{\dot{}}{⨣}Fn\left(B\times B\right)$$