Theorem frmdadd

Description: Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015)

	Ref	Expression
Hypotheses	frmdbas.m	⊢ 𝑀 = ( freeMnd ‘ 𝐼 )
	frmdbas.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
	frmdplusg.p	⊢ + = ( +g ‘ 𝑀 )
Assertion	frmdadd	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ++ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		frmdbas.m	⊢ 𝑀 = ( freeMnd ‘ 𝐼 )
2		frmdbas.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
3		frmdplusg.p	⊢ + = ( +g ‘ 𝑀 )
4	1 2 3	frmdplusg	⊢ + = ( ++ ↾ ( 𝐵 × 𝐵 ) )
5	4	oveqi	⊢ ( 𝑋 + 𝑌 ) = ( 𝑋 ( ++ ↾ ( 𝐵 × 𝐵 ) ) 𝑌 )
6		ovres	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( ++ ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 ++ 𝑌 ) )
7	5 6	eqtrid	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ++ 𝑌 ) )