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Metamath Proof Explorer

Theorem mndass

Description: A monoid operation is associative. (Contributed by NM, 14-Aug-2011) (Proof shortened by AV, 8-Feb-2020)

Ref Expression
Hypotheses mndcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
mndcl.p ⊢ + = ( +g ‘ 𝐺 )
Assertion mndass ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 mndcl.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 mndcl.p ⊢ + = ( +g ‘ 𝐺 )
3 mndsgrp ⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp )
4 1 2 sgrpass ⊢ ( ( 𝐺 ∈ Smgrp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
5 3 4 sylan ⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )