Theorem caovdir

Description: Reverse distributive law. (Contributed by NM, 26-Aug-1995)

	Ref	Expression
Hypotheses	caovdir.1	⊢ 𝐴 ∈ V
	caovdir.2	⊢ 𝐵 ∈ V
	caovdir.3	⊢ 𝐶 ∈ V
	caovdir.com	⊢ ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 )
	caovdir.distr	⊢ ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) )
Assertion	caovdir	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		caovdir.1	⊢ 𝐴 ∈ V
2		caovdir.2	⊢ 𝐵 ∈ V
3		caovdir.3	⊢ 𝐶 ∈ V
4		caovdir.com	⊢ ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 )
5		caovdir.distr	⊢ ( 𝑥 𝐺 ( 𝑦 𝐹 𝑧 ) ) = ( ( 𝑥 𝐺 𝑦 ) 𝐹 ( 𝑥 𝐺 𝑧 ) )
6	3 1 2 5	caovdi	⊢ ( 𝐶 𝐺 ( 𝐴 𝐹 𝐵 ) ) = ( ( 𝐶 𝐺 𝐴 ) 𝐹 ( 𝐶 𝐺 𝐵 ) )
7		ovex	⊢ ( 𝐴 𝐹 𝐵 ) ∈ V
8	3 7 4	caovcom	⊢ ( 𝐶 𝐺 ( 𝐴 𝐹 𝐵 ) ) = ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 )
9	3 1 4	caovcom	⊢ ( 𝐶 𝐺 𝐴 ) = ( 𝐴 𝐺 𝐶 )
10	3 2 4	caovcom	⊢ ( 𝐶 𝐺 𝐵 ) = ( 𝐵 𝐺 𝐶 )
11	9 10	oveq12i	⊢ ( ( 𝐶 𝐺 𝐴 ) 𝐹 ( 𝐶 𝐺 𝐵 ) ) = ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐶 ) )
12	6 8 11	3eqtr3i	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐹 ( 𝐵 𝐺 𝐶 ) )