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Metamath Proof Explorer

Theorem caov411d

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 caov411d ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 ( 𝐴 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 1 3 4 5 6 7 caov4d ⊢ ( 𝜑 → ( ( 𝐵 𝐹 𝐴 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐵 𝐹 𝐶 ) 𝐹 ( 𝐴 𝐹 𝐷 ) ) )
9 4 2 1 caovcomd ⊢ ( 𝜑 → ( 𝐵 𝐹 𝐴 ) = ( 𝐴 𝐹 𝐵 ) )
10 9 oveq1d ⊢ ( 𝜑 → ( ( 𝐵 𝐹 𝐴 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) )
11 4 2 3 caovcomd ⊢ ( 𝜑 → ( 𝐵 𝐹 𝐶 ) = ( 𝐶 𝐹 𝐵 ) )
12 11 oveq1d ⊢ ( 𝜑 → ( ( 𝐵 𝐹 𝐶 ) 𝐹 ( 𝐴 𝐹 𝐷 ) ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 ( 𝐴 𝐹 𝐷 ) ) )
13 8 10 12 3eqtr3d ⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 ( 𝐴 𝐹 𝐷 ) ) )