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Theorem iscllaw 47329
Description: The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
Assertion
Ref Expression
iscllaw (( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (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.
StepHypRef Expression
1 simpr 483 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 7432 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
32adantr 479 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 𝑦))
43, 1eleq12d 2823 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 𝑦) ∈ 𝑀))
51, 4raleqbidv 3340 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
61, 5raleqbidv 3340 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
7 df-cllaw 47326 . 2 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
86, 7brabga 5540 1 (( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058   class class class wbr 5152  (class class class)co 7426   clLaw ccllaw 47323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-iota 6505  df-fv 6561  df-ov 7429  df-cllaw 47326
This theorem is referenced by:  clcllaw  47331  mgmplusgiopALT  47334  clintopcllaw  47351  mgm2mgm  47367
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