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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 ⊢ + = ( +g ‘ 𝐺 )
mndlrid.o ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion mndlrid ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )

Proof

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