Theorem caov4d

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
	caovd.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
	caovd.com	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
	caovd.ass	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
	caovd.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑆 )
	caovd.cl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝑆 )
Assertion	caov4d	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
2		caovd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
3		caovd.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
4		caovd.com	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
5		caovd.ass	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
6		caovd.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑆 )
7		caovd.cl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝑆 )
8	2 3 6 4 5	caov12d	⊢ ( 𝜑 → ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) )
9	8	oveq2d	⊢ ( 𝜑 → ( 𝐴 𝐹 ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) ) )
10	7 3 6	caovcld	⊢ ( 𝜑 → ( 𝐶 𝐹 𝐷 ) ∈ 𝑆 )
11	5 1 2 10	caovassd	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( 𝐴 𝐹 ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) ) )
12	7 2 6	caovcld	⊢ ( 𝜑 → ( 𝐵 𝐹 𝐷 ) ∈ 𝑆 )
13	5 1 3 12	caovassd	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) ) )
14	9 11 13	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) ) )