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Theorem assintopcllaw 42173
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 6259 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopval 42166 . . . . 5 (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
32eleq2d 2716 . . . 4 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
4 breq1 4688 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3396 . . . 4 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
63, 5syl6bb 276 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
7 clintopcllaw 42172 . . . 4 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
87adantr 480 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀) → clLaw 𝑀)
96, 8syl6bi 243 . 2 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) → clLaw 𝑀))
101, 9mpcom 38 1 ( ∈ ( assIntOp ‘𝑀) → clLaw 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  {crab 2945  Vcvv 3231   class class class wbr 4685  cfv 5926   clLaw ccllaw 42144   assLaw casslaw 42145   clIntOp cclintop 42158   assIntOp cassintop 42159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-cllaw 42147  df-intop 42160  df-clintop 42161  df-assintop 42162
This theorem is referenced by: (None)
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