Theorem iscllaw 46589

Description: The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)

Assertion

Ref	Expression
iscllaw	⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))

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

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

Proof of Theorem iscllaw

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

Step	Hyp	Ref	Expression
1		simpr 485	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
2		oveq 7414	. . . . . 6 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
3	2	adantr 481	. . . . 5 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
4	3, 1	eleq12d 2827	. . . 4 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 ⚬ 𝑦) ∈ 𝑀))
5	1, 4	raleqbidv 3342	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
6	1, 5	raleqbidv 3342	. 2 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
7		df-cllaw 46586	. 2 ⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
8	6, 7	brabga 5534	1 ⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   class class class wbr 5148  (class class class)co 7408   clLaw ccllaw 46583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7411  df-cllaw 46586

This theorem is referenced by:  clcllaw  46591  mgmplusgiopALT  46594  clintopcllaw  46611  mgm2mgm  46627