Theorem oppcco

Description: Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	oppcco.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	oppcco.c	⊢ · = ( comp ‘ 𝐶 )
	oppcco.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
	oppcco.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	oppcco.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	oppcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	oppcco	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝑂 ) 𝑍 ) 𝐹 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcco.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		oppcco.c	⊢ · = ( comp ‘ 𝐶 )
3		oppcco.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
4		oppcco.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		oppcco.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		oppcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7	1 2 3 4 5 6	oppccofval	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝑂 ) 𝑍 ) = tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) )
8	7	oveqd	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝑂 ) 𝑍 ) 𝐹 ) = ( 𝐺 tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) )
9		ovtpos	⊢ ( 𝐺 tpos ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) 𝐺 )
10	8 9	eqtrdi	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝑂 ) 𝑍 ) 𝐹 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑌 ⟩ · 𝑋 ) 𝐺 ) )