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Theorem assintopasslaw 45368
Description: The associative low holds for a associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
Assertion
Ref Expression
assintopasslaw ( ∈ ( assIntOp ‘𝑀) → assLaw 𝑀)

Proof of Theorem assintopasslaw
StepHypRef Expression
1 assintop 45364 . 2 ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
21simprd 496 1 ( ∈ ( assIntOp ‘𝑀) → assLaw 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110   class class class wbr 5079   × cxp 5587  wf 6427  cfv 6431   assLaw casslaw 45339   assIntOp cassintop 45353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7818  df-2nd 7819  df-map 8592  df-intop 45354  df-clintop 45355  df-assintop 45356
This theorem is referenced by:  assintopass  45369
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