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Theorem iscllaw 46209
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 486 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 7364 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
32adantr 482 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 𝑦))
43, 1eleq12d 2828 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 𝑦) ∈ 𝑀))
51, 4raleqbidv 3318 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
61, 5raleqbidv 3318 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
7 df-cllaw 46206 . 2 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
86, 7brabga 5492 1 (( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061   class class class wbr 5106  (class class class)co 7358   clLaw ccllaw 46203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-iota 6449  df-fv 6505  df-ov 7361  df-cllaw 46206
This theorem is referenced by:  clcllaw  46211  mgmplusgiopALT  46214  clintopcllaw  46231  mgm2mgm  46247
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