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Theorem assintopass 47387
Description: An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
Assertion
Ref Expression
assintopass ( ∈ ( assIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem assintopass
StepHypRef Expression
1 id 22 . 2 ( ∈ ( assIntOp ‘𝑀) → ∈ ( assIntOp ‘𝑀))
2 elfvex 6929 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
3 assintopasslaw 47386 . . 3 ( ∈ ( assIntOp ‘𝑀) → assLaw 𝑀)
4 isasslaw 47365 . . 3 (( ∈ ( assIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
53, 4syl5ibcom 244 . 2 ( ∈ ( assIntOp ‘𝑀) → (( ∈ ( assIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
61, 2, 5mp2and 697 1 ( ∈ ( assIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3051  Vcvv 3463   class class class wbr 5143  cfv 6542  (class class class)co 7415   assLaw casslaw 47357   assIntOp cassintop 47371
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 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843  df-asslaw 47361  df-intop 47372  df-clintop 47373  df-assintop 47374
This theorem is referenced by: (None)
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