Theorem caovdig

Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995) (Revised by Mario Carneiro, 26-Jul-2014)

	Ref	Expression
Hypothesis	caovdig.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐻 ( 𝑥 𝐺 𝑧 ) ) )
Assertion	caovdig	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 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	1	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐾 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐻 ( 𝑥 𝐺 𝑧 ) ) )
3		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( 𝐴 𝐺 ( 𝑦 𝐹 𝑧 ) ) )
4		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 𝑦 ) = ( 𝐴 𝐺 𝑦 ) )
5		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐺 𝑧 ) = ( 𝐴 𝐺 𝑧 ) )
6	4 5	oveq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐺 𝑦 ) 𝐻 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝐴 𝐺 𝑦 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) )
7	3 6	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐻 ( 𝑥 𝐺 𝑧 ) ) ↔ ( 𝐴 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝐴 𝐺 𝑦 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) ) )
8		oveq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐹 𝑧 ) = ( 𝐵 𝐹 𝑧 ) )
9	8	oveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( 𝐴 𝐺 ( 𝐵 𝐹 𝑧 ) ) )
10		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐺 𝑦 ) = ( 𝐴 𝐺 𝐵 ) )
11	10	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐺 𝑦 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) )
12	9 11	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝐴 𝐺 𝑦 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) ↔ ( 𝐴 𝐺 ( 𝐵 𝐹 𝑧 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) ) )
13		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝐵 𝐹 𝑧 ) = ( 𝐵 𝐹 𝐶 ) )
14	13	oveq2d	⊢ ( 𝑧 = 𝐶 → ( 𝐴 𝐺 ( 𝐵 𝐹 𝑧 ) ) = ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) )
15		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝐴 𝐺 𝑧 ) = ( 𝐴 𝐺 𝐶 ) )
16	15	oveq2d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝐶 ) ) )
17	14 16	eqeq12d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐺 ( 𝐵 𝐹 𝑧 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝑧 ) ) ↔ ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝐶 ) ) ) )
18	7 12 17	rspc3v	⊢ ( ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝐾 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐻 ( 𝑥 𝐺 𝑧 ) ) → ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝐶 ) ) ) )
19	2 18	mpan9	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 𝐴 𝐺 ( 𝐵 𝐹 𝐶 ) ) = ( ( 𝐴 𝐺 𝐵 ) 𝐻 ( 𝐴 𝐺 𝐶 ) ) )