Metamath Proof Explorer


Theorem mndlrid

Description: A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011)

Ref Expression
Hypotheses mndlrid.b 𝐵 = ( Base ‘ 𝐺 )
mndlrid.p + = ( +g𝐺 )
mndlrid.o 0 = ( 0g𝐺 )
Assertion mndlrid ( ( 𝐺 ∈ Mnd ∧ 𝑋𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 mndlrid.b 𝐵 = ( Base ‘ 𝐺 )
2 mndlrid.p + = ( +g𝐺 )
3 mndlrid.o 0 = ( 0g𝐺 )
4 1 2 mndid ( 𝐺 ∈ Mnd → ∃ 𝑦𝐵𝑥𝐵 ( ( 𝑦 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑦 ) = 𝑥 ) )
5 1 3 2 4 mgmlrid ( ( 𝐺 ∈ Mnd ∧ 𝑋𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )