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Metamath Proof Explorer

Theorem grprid

Description: The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011)

Ref Expression
Hypotheses grpbn0.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grplid.p ⊢ + = ( +g ‘ 𝐺 )
grplid.o ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion grprid ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + 0 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 grpbn0.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grplid.p ⊢ + = ( +g ‘ 𝐺 )
3 grplid.o ⊢ 0 = ( 0g ‘ 𝐺 )
4 grpmnd ⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
5 1 2 3 mndrid ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + 0 ) = 𝑋 )
6 4 5 sylan ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + 0 ) = 𝑋 )