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Metamath Proof Explorer

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 )