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Metamath Proof Explorer

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 ) ) )