Theorem iscomlaw 48772

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

Assertion

Ref	Expression
iscomlaw	⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))

Distinct variable groups:   𝑥,𝑀,𝑦   𝑥, ⚬ ,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem iscomlaw

Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
2		oveq 7396	. . . . . 6 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
3		oveq 7396	. . . . . 6 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑥) = (𝑦 ⚬ 𝑥))
4	2, 3	eqeq12d 2777	. . . . 5 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))
5	4	adantr 484	. . . 4 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))
6	1, 5	raleqbidv 3335	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))
7	1, 6	raleqbidv 3335	. 2 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))
8		df-comlaw 48769	. 2 ⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
9	7, 8	brabga 5501	1 ⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5097  (class class class)co 7390   comLaw ccomlaw 48767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-iota 6471  df-fv 6523  df-ov 7393  df-comlaw 48769

This theorem is referenced by: (None)