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Theorem isassintop 41636
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 41632 . . . . 5 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
21eleq2d 2673 . . . 4 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3 breq1 4586 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
43elrab 3331 . . . 4 ( ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀))
52, 4syl6bb 275 . . 3 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)))
6 elmapi 7765 . . . . . 6 ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76ad2antrl 760 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
8 isasslaw 41618 . . . . . . . 8 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
98biimpd 218 . . . . . . 7 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
109impancom 455 . . . . . 6 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀) → (𝑀𝑉 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1110impcom 445 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
127, 11jca 553 . . . 4 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1312ex 449 . . 3 (𝑀𝑉 → (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
145, 13sylbid 229 . 2 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
15 isclintop 41633 . . . . . . 7 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
1615biimprcd 239 . . . . . 6 ( :(𝑀 × 𝑀)⟶𝑀 → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1716adantr 480 . . . . 5 (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1817impcom 445 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( clIntOp ‘𝑀))
19 sqxpexg 6861 . . . . . . . . 9 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
20 fex 6394 . . . . . . . . 9 (( :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ∈ V)
2119, 20sylan2 490 . . . . . . . 8 (( :(𝑀 × 𝑀)⟶𝑀𝑀𝑉) → ∈ V)
2221ancoms 468 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → ∈ V)
23 simpl 472 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → 𝑀𝑉)
24 isasslaw 41618 . . . . . . . 8 (( ∈ V ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2524bicomd 212 . . . . . . 7 (( ∈ V ∧ 𝑀𝑉) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2622, 23, 25syl2anc 691 . . . . . 6 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2726biimpd 218 . . . . 5 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → assLaw 𝑀))
2827impr 647 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → assLaw 𝑀)
29 assintopval 41631 . . . . . . 7 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3029adantr 480 . . . . . 6 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3130eleq2d 2673 . . . . 5 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
323elrab 3331 . . . . 5 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
3331, 32syl6bb 275 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
3418, 28, 33mpbir2and 959 . . 3 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( assIntOp ‘𝑀))
3534ex 449 . 2 (𝑀𝑉 → (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → ∈ ( assIntOp ‘𝑀)))
3614, 35impbid 201 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744   assLaw casslaw 41610   clIntOp cclintop 41623   assIntOp cassintop 41624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-asslaw 41614  df-intop 41625  df-clintop 41626  df-assintop 41627
This theorem is referenced by: (None)
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