Metamath Proof Explorer


Theorem caovdirg

Description: Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014)

Ref Expression
Hypothesis caovdirg.1 ( ( 𝜑 ∧ ( 𝑥𝑆𝑦𝑆𝑧𝐾 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐺 𝑧 ) = ( ( 𝑥 𝐺 𝑧 ) 𝐻 ( 𝑦 𝐺 𝑧 ) ) )
Assertion caovdirg ( ( 𝜑 ∧ ( 𝐴𝑆𝐵𝑆𝐶𝐾 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐻 ( 𝐵 𝐺 𝐶 ) ) )

Proof

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