Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  assintopcllaw Structured version   Visualization version   GIF version

Theorem assintopcllaw 41619
Description: The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
Assertion
Ref Expression
assintopcllaw ( ∈ ( assIntOp ‘𝑀) → clLaw 𝑀)

Proof of Theorem assintopcllaw
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6115 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopval 41612 . . . . 5 (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
32eleq2d 2672 . . . 4 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
4 breq1 4580 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3330 . . . 4 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
63, 5syl6bb 274 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
7 clintopcllaw 41618 . . . 4 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
87adantr 479 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀) → clLaw 𝑀)
96, 8syl6bi 241 . 2 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) → clLaw 𝑀))
101, 9mpcom 37 1 ( ∈ ( assIntOp ‘𝑀) → clLaw 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  {crab 2899  Vcvv 3172   class class class wbr 4577  cfv 5789   clLaw ccllaw 41590   assLaw casslaw 41591   clIntOp cclintop 41604   assIntOp cassintop 41605
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-8 1978  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-pow 4763  ax-pr 4827  ax-un 6824
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-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-map 7723  df-cllaw 41593  df-intop 41606  df-clintop 41607  df-assintop 41608
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator