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Theorem assintopass 42519
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 6409 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
3 assintopasslaw 42518 . . 3 ( ∈ ( assIntOp ‘𝑀) → assLaw 𝑀)
4 isasslaw 42497 . . 3 (( ∈ ( assIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
53, 4syl5ibcom 236 . 2 ( ∈ ( assIntOp ‘𝑀) → (( ∈ ( assIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
61, 2, 5mp2and 690 1 ( ∈ ( assIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350   class class class wbr 4809  cfv 6068  (class class class)co 6842   assLaw casslaw 42489   assIntOp cassintop 42503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-asslaw 42493  df-intop 42504  df-clintop 42505  df-assintop 42506
This theorem is referenced by: (None)
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