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Theorem assintopcllaw 47382
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 6928 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopval 47375 . . . . 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 5147 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3676 . . . 4 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
63, 5bitrdi 286 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
7 clintopcllaw 47381 . . . 4 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
87adantr 479 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀) → clLaw 𝑀)
96, 8biimtrdi 252 . 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 394  wcel 2098  {crab 3419  Vcvv 3463   class class class wbr 5144  cfv 6543   clLaw ccllaw 47353   assLaw casslaw 47354   clIntOp cclintop 47367   assIntOp cassintop 47368
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 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8840  df-cllaw 47356  df-intop 47369  df-clintop 47370  df-assintop 47371
This theorem is referenced by: (None)
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