Theorem iscllaw 49104

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 490	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
2		oveq 7419	. . . . . 6 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
3	2	adantr 486	. . . . 5 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
4	3, 1	eleq12d 2854	. . . 4 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 ⚬ 𝑦) ∈ 𝑀))
5	1, 4	raleqbidv 3334	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
6	1, 5	raleqbidv 3334	. 2 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
7		df-cllaw 49101	. 2 ⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
8	6, 7	brabga 5512	1 ⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   class class class wbr 5103  (class class class)co 7413   clLaw ccllaw 49098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6489  df-fv 6541  df-ov 7416  df-cllaw 49101

This theorem is used by:  clcllaw  49106  mgmplusgiopALT  49109  clintopcllaw  49126  mgm2mgm  49142