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Theorem iscllaw 42410
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 473 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 6889 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
32adantr 468 . . . . 5 ((𝑜 = 𝑚 = 𝑀) → (𝑥𝑜𝑦) = (𝑥 𝑦))
43, 1eleq12d 2890 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) ∈ 𝑚 ↔ (𝑥 𝑦) ∈ 𝑀))
51, 4raleqbidv 3352 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
61, 5raleqbidv 3352 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
7 df-cllaw 42407 . 2 clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
86, 7brabga 5197 1 (( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wral 3107   class class class wbr 4855  (class class class)co 6883   clLaw ccllaw 42404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-iota 6073  df-fv 6118  df-ov 6886  df-cllaw 42407
This theorem is referenced by:  clcllaw  42412  mgmplusgiopALT  42415  clintopcllaw  42432  mgm2mgm  42448
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