Metamath Proof Explorer


Theorem iscomlaw

Description: The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020)

Ref Expression
Assertion iscomlaw ( ( 𝑉𝑀𝑊 ) → ( comLaw 𝑀 ↔ ∀ 𝑥𝑀𝑦𝑀 ( 𝑥 𝑦 ) = ( 𝑦 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 simpr ( ( 𝑜 = 𝑚 = 𝑀 ) → 𝑚 = 𝑀 )
2 oveq ( 𝑜 = → ( 𝑥 𝑜 𝑦 ) = ( 𝑥 𝑦 ) )
3 oveq ( 𝑜 = → ( 𝑦 𝑜 𝑥 ) = ( 𝑦 𝑥 ) )
4 2 3 eqeq12d ( 𝑜 = → ( ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ( 𝑥 𝑦 ) = ( 𝑦 𝑥 ) ) )
5 4 adantr ( ( 𝑜 = 𝑚 = 𝑀 ) → ( ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ( 𝑥 𝑦 ) = ( 𝑦 𝑥 ) ) )
6 1 5 raleqbidv ( ( 𝑜 = 𝑚 = 𝑀 ) → ( ∀ 𝑦𝑚 ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ∀ 𝑦𝑀 ( 𝑥 𝑦 ) = ( 𝑦 𝑥 ) ) )
7 1 6 raleqbidv ( ( 𝑜 = 𝑚 = 𝑀 ) → ( ∀ 𝑥𝑚𝑦𝑚 ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) ↔ ∀ 𝑥𝑀𝑦𝑀 ( 𝑥 𝑦 ) = ( 𝑦 𝑥 ) ) )
8 df-comlaw comLaw = { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥𝑚𝑦𝑚 ( 𝑥 𝑜 𝑦 ) = ( 𝑦 𝑜 𝑥 ) }
9 7 8 brabga ( ( 𝑉𝑀𝑊 ) → ( comLaw 𝑀 ↔ ∀ 𝑥𝑀𝑦𝑀 ( 𝑥 𝑦 ) = ( 𝑦 𝑥 ) ) )