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Metamath Proof Explorer

Theorem caovord2d

Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovordg.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
caovordd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
caovordd.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
caovordd.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
caovord2d.com ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
Assertion caovord2d ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐴 𝐹 𝐶 ) 𝑅 ( 𝐵 𝐹 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 caovordg.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
2 caovordd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3 caovordd.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4 caovordd.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
5 caovord2d.com ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
6 1 2 3 4 caovordd ⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
7 5 4 2 caovcomd ⊢ ( 𝜑 → ( 𝐶 𝐹 𝐴 ) = ( 𝐴 𝐹 𝐶 ) )
8 5 4 3 caovcomd ⊢ ( 𝜑 → ( 𝐶 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐶 ) )
9 7 8 breq12d ⊢ ( 𝜑 → ( ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ↔ ( 𝐴 𝐹 𝐶 ) 𝑅 ( 𝐵 𝐹 𝐶 ) ) )
10 6 9 bitrd ⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐴 𝐹 𝐶 ) 𝑅 ( 𝐵 𝐹 𝐶 ) ) )