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Metamath Proof Explorer

Theorem mndlid

Description: The identity element of a monoid is a left 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 mndlid ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 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 3 mndlrid ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( ( 0 + 𝑋 ) = 𝑋 ∧ ( 𝑋 + 0 ) = 𝑋 ) )
5 4 simpld ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 )