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Metamath Proof Explorer

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 ) ) )