Theorem mndtcco2

Description: The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024)

	Ref	Expression
Hypotheses	mndtcbas.c	⊢ ( 𝜑 → 𝐶 = ( MndToCat ‘ 𝑀 ) )
	mndtcbas.m	⊢ ( 𝜑 → 𝑀 ∈ Mnd )
	mndtcbas.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
	mndtchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	mndtchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	mndtcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	mndtcco.o	⊢ ( 𝜑 → · = ( comp ‘ 𝐶 ) )
	mndtcco2.o2	⊢ ( 𝜑 → ⚬ = ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) )
Assertion	mndtcco2	⊢ ( 𝜑 → ( 𝐺 ⚬ 𝐹 ) = ( 𝐺 ( +g ‘ 𝑀 ) 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		mndtcbas.c	⊢ ( 𝜑 → 𝐶 = ( MndToCat ‘ 𝑀 ) )
2		mndtcbas.m	⊢ ( 𝜑 → 𝑀 ∈ Mnd )
3		mndtcbas.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
4		mndtchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		mndtchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		mndtcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7		mndtcco.o	⊢ ( 𝜑 → · = ( comp ‘ 𝐶 ) )
8		mndtcco2.o2	⊢ ( 𝜑 → ⚬ = ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) )
9	1 2 3 4 5 6 7	mndtcco	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) = ( +g ‘ 𝑀 ) )
10	8 9	eqtrd	⊢ ( 𝜑 → ⚬ = ( +g ‘ 𝑀 ) )
11	10	oveqd	⊢ ( 𝜑 → ( 𝐺 ⚬ 𝐹 ) = ( 𝐺 ( +g ‘ 𝑀 ) 𝐹 ) )