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Theorem iscllaw 41596
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 475 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 6532 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
32adantr 479 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 𝑦))
43, 1eleq12d 2681 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 𝑦) ∈ 𝑀))
51, 4raleqbidv 3128 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
61, 5raleqbidv 3128 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
7 df-cllaw 41593 . 2 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
86, 7brabga 4903 1 (( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895   class class class wbr 4577  (class class class)co 6526   clLaw ccllaw 41590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-iota 5753  df-fv 5797  df-ov 6529  df-cllaw 41593
This theorem is referenced by:  clcllaw  41598  mgmplusgiopALT  41601  clintopcllaw  41618  mgm2mgm  41634
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