Theorem mndplusf

Description: The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015) (Proof shortened by AV, 3-Feb-2020)

	Ref	Expression
Hypotheses	mndplusf.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mndplusf.2	⊢ ⨣ = ( +𝑓 ‘ 𝐺 )
Assertion	mndplusf	⊢ ( 𝐺 ∈ Mnd → ⨣ : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mndplusf.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mndplusf.2	⊢ ⨣ = ( +𝑓 ‘ 𝐺 )
3		mndmgm	⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Mgm )
4	1 2	mgmplusf	⊢ ( 𝐺 ∈ Mgm → ⨣ : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )
5	3 4	syl	⊢ ( 𝐺 ∈ Mnd → ⨣ : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )