Theorem caovdird

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

	Ref	Expression
Hypotheses	caovdirg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) )
	caovdird.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovdird.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
	caovdird.4	⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
Assertion	caovdird	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovdirg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) )
2		caovdird.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3		caovdird.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4		caovdird.4	⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
5		id	⊢ ( 𝜑 → 𝜑 )
6	1	caovdirg	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) )
7	5 2 3 4 6	syl13anc	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) )