Theorem grpplusf

Description: The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Hypotheses	grpplusf.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpplusf.2	⊢ 𝐹 = ( +𝑓 ‘ 𝐺 )
Assertion	grpplusf	⊢ ( 𝐺 ∈ Grp → 𝐹 : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpplusf.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpplusf.2	⊢ 𝐹 = ( +𝑓 ‘ 𝐺 )
3		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
4	1 2	mndplusf	⊢ ( 𝐺 ∈ Mnd → 𝐹 : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )
5	3 4	syl	⊢ ( 𝐺 ∈ Grp → 𝐹 : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )