Theorem iscllaw 43461

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 477	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
2		oveq 6984	. . . . . 6 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
3	2	adantr 473	. . . . 5 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
4	3, 1	eleq12d 2860	. . . 4 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 ⚬ 𝑦) ∈ 𝑀))
5	1, 4	raleqbidv 3341	. . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
6	1, 5	raleqbidv 3341	. 2 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
7		df-cllaw 43458	. 2 ⊢ clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
8	6, 7	brabga 5276	1 ⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3088   class class class wbr 4930  (class class class)co 6978   clLaw ccllaw 43455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-iota 6154  df-fv 6198  df-ov 6981  df-cllaw 43458

This theorem is referenced by:  clcllaw  43463  mgmplusgiopALT  43466  clintopcllaw  43483  mgm2mgm  43499