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Theorem assintopcllaw 47100
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 6920 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopval 47093 . . . . 5 (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
32eleq2d 2811 . . . 4 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
4 breq1 5142 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3676 . . . 4 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
63, 5bitrdi 287 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
7 clintopcllaw 47099 . . . 4 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
87adantr 480 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀) → clLaw 𝑀)
96, 8syl6bi 253 . 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 395  wcel 2098  {crab 3424  Vcvv 3466   class class class wbr 5139  cfv 6534   clLaw ccllaw 47071   assLaw casslaw 47072   clIntOp cclintop 47085   assIntOp cassintop 47086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-cllaw 47074  df-intop 47087  df-clintop 47088  df-assintop 47089
This theorem is referenced by: (None)
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