Theorem caovdir2d

Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovdir2d.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) ) )
	caovdir2d.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovdir2d.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
	caovdir2d.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
	caovdir2d.cl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝑆 )
	caovdir2d.com	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 ) )
Assertion	caovdir2d	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovdir2d.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) ) )
2		caovdir2d.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3		caovdir2d.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4		caovdir2d.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
5		caovdir2d.cl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝑆 )
6		caovdir2d.com	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 ) )
7	1 4 2 3	caovdid	⊢ ( 𝜑 → ( 𝐶 𝐺 ( 𝐴 𝐹 𝐵 ) ) = ( ( 𝐶 𝐺 𝐴 ) 𝐹 ( 𝐶 𝐺 𝐵 ) ) )
8	5 2 3	caovcld	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝑆 )
9	6 8 4	caovcomd	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( 𝐶 𝐺 ( 𝐴 𝐹 𝐵 ) ) )
10	6 2 4	caovcomd	⊢ ( 𝜑 → ( 𝐴 𝐺 𝐶 ) = ( 𝐶 𝐺 𝐴 ) )
11	6 3 4	caovcomd	⊢ ( 𝜑 → ( 𝐵 𝐺 𝐶 ) = ( 𝐶 𝐺 𝐵 ) )
12	10 11	oveq12d	⊢ ( 𝜑 → ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐶 𝐺 𝐴 ) 𝐹 ( 𝐶 𝐺 𝐵 ) ) )
13	7 9 12	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐶 ) ) )