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 + = ( +g𝐺 )
grplid.o 0 = ( 0g𝐺 )
Assertion grprid ( ( 𝐺 ∈ Grp ∧ 𝑋𝐵 ) → ( 𝑋 + 0 ) = 𝑋 )

Proof

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