Theorem grpass

Description: A group operation is associative. (Contributed by NM, 14-Aug-2011)

	Ref	Expression
Hypotheses	grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpcl.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	grpass	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpcl.p	⊢ + = ( +g ‘ 𝐺 )
3		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
4	1 2	mndass	⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )
5	3 4	sylan	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) )