Theorem caovassg

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013) (Revised by Mario Carneiro, 26-May-2014)

	Ref	Expression
Hypothesis	caovassg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
Assertion	caovassg	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovassg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
2	1	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
3		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) )
4	3	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( ( 𝐴 𝐹 𝑦 ) 𝐹 𝑧 ) )
5		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) = ( 𝐴 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
6	4 5	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) ↔ ( ( 𝐴 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝐴 𝐹 ( 𝑦 𝐹 𝑧 ) ) ) )
7		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) )
8	7	oveq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) 𝐹 𝑧 ) = ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝑧 ) )
9		oveq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐹 𝑧 ) = ( 𝐵 𝐹 𝑧 ) )
10	9	oveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 ( 𝑦 𝐹 𝑧 ) ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝑧 ) ) )
11	8 10	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝐴 𝐹 ( 𝑦 𝐹 𝑧 ) ) ↔ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝑧 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝑧 ) ) ) )
12		oveq2	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝑧 ) = ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) )
13		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝐵 𝐹 𝑧 ) = ( 𝐵 𝐹 𝐶 ) )
14	13	oveq2d	⊢ ( 𝑧 = 𝐶 → ( 𝐴 𝐹 ( 𝐵 𝐹 𝑧 ) ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )
15	12 14	eqeq12d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝑧 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝑧 ) ) ↔ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) ) )
16	6 11 15	rspc3v	⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) ) )
17	2 16	mpan9	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )