Metamath Proof Explorer


Theorem mndid

Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011)

Ref Expression
Hypotheses mndcl.b 𝐵 = ( Base ‘ 𝐺 )
mndcl.p + = ( +g𝐺 )
Assertion mndid ( 𝐺 ∈ Mnd → ∃ 𝑢𝐵𝑥𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )

Proof

Step Hyp Ref Expression
1 mndcl.b 𝐵 = ( Base ‘ 𝐺 )
2 mndcl.p + = ( +g𝐺 )
3 1 2 ismnd ( 𝐺 ∈ Mnd ↔ ( ∀ 𝑥𝐵𝑦𝐵 ( ( 𝑥 + 𝑦 ) ∈ 𝐵 ∧ ∀ 𝑧𝐵 ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ∧ ∃ 𝑢𝐵𝑥𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ) )
4 3 simprbi ( 𝐺 ∈ Mnd → ∃ 𝑢𝐵𝑥𝐵 ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) )