Theorem caovdid

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

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

Proof

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