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Theorem isassintop 44110
Description: The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
Assertion
Ref Expression
isassintop (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem isassintop
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 assintopmap 44106 . . . . 5 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
21eleq2d 2898 . . . 4 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3 breq1 5062 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
43elrab 3680 . . . 4 ( ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀))
52, 4syl6bb 289 . . 3 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀)))
6 elmapi 8422 . . . . . 6 ( ∈ (𝑀m (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76ad2antrl 726 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
8 isasslaw 44092 . . . . . . . 8 (( ∈ (𝑀m (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
98biimpd 231 . . . . . . 7 (( ∈ (𝑀m (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
109impancom 454 . . . . . 6 (( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀) → (𝑀𝑉 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1110impcom 410 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
127, 11jca 514 . . . 4 ((𝑀𝑉 ∧ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1312ex 415 . . 3 (𝑀𝑉 → (( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
145, 13sylbid 242 . 2 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
15 isclintop 44107 . . . . . . 7 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
1615biimprcd 252 . . . . . 6 ( :(𝑀 × 𝑀)⟶𝑀 → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1716adantr 483 . . . . 5 (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1817impcom 410 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( clIntOp ‘𝑀))
19 sqxpexg 7471 . . . . . . . . 9 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
20 fex 6983 . . . . . . . . 9 (( :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ∈ V)
2119, 20sylan2 594 . . . . . . . 8 (( :(𝑀 × 𝑀)⟶𝑀𝑀𝑉) → ∈ V)
2221ancoms 461 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → ∈ V)
23 simpl 485 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → 𝑀𝑉)
24 isasslaw 44092 . . . . . . . 8 (( ∈ V ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2524bicomd 225 . . . . . . 7 (( ∈ V ∧ 𝑀𝑉) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2622, 23, 25syl2anc 586 . . . . . 6 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2726biimpd 231 . . . . 5 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → assLaw 𝑀))
2827impr 457 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → assLaw 𝑀)
29 assintopval 44105 . . . . . . 7 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3029adantr 483 . . . . . 6 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3130eleq2d 2898 . . . . 5 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
323elrab 3680 . . . . 5 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
3331, 32syl6bb 289 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
3418, 28, 33mpbir2and 711 . . 3 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( assIntOp ‘𝑀))
3534ex 415 . 2 (𝑀𝑉 → (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → ∈ ( assIntOp ‘𝑀)))
3614, 35impbid 214 1 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  {crab 3142  Vcvv 3495   class class class wbr 5059   × cxp 5548  wf 6346  cfv 6350  (class class class)co 7150  m cmap 8400   assLaw casslaw 44084   clIntOp cclintop 44097   assIntOp cassintop 44098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-asslaw 44088  df-intop 44099  df-clintop 44100  df-assintop 44101
This theorem is referenced by: (None)
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