Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isassintop Structured version   Visualization version   GIF version

Theorem isassintop 41631
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 41627 . . . . 5 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
21eleq2d 2672 . . . 4 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3 breq1 4580 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
43elrab 3330 . . . 4 ( ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀))
52, 4syl6bb 274 . . 3 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)))
6 elmapi 7742 . . . . . 6 ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76ad2antrl 759 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
8 isasslaw 41613 . . . . . . . 8 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
98biimpd 217 . . . . . . 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 444 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
127, 11jca 552 . . . 4 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1312ex 448 . . 3 (𝑀𝑉 → (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
145, 13sylbid 228 . 2 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
15 isclintop 41628 . . . . . . 7 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
1615biimprcd 238 . . . . . 6 ( :(𝑀 × 𝑀)⟶𝑀 → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1716adantr 479 . . . . 5 (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1817impcom 444 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( clIntOp ‘𝑀))
19 sqxpexg 6838 . . . . . . . . 9 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
20 fex 6372 . . . . . . . . 9 (( :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ∈ V)
2119, 20sylan2 489 . . . . . . . 8 (( :(𝑀 × 𝑀)⟶𝑀𝑀𝑉) → ∈ V)
2221ancoms 467 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → ∈ V)
23 simpl 471 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → 𝑀𝑉)
24 isasslaw 41613 . . . . . . . 8 (( ∈ V ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2524bicomd 211 . . . . . . 7 (( ∈ V ∧ 𝑀𝑉) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2622, 23, 25syl2anc 690 . . . . . 6 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2726biimpd 217 . . . . 5 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → assLaw 𝑀))
2827impr 646 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → assLaw 𝑀)
29 assintopval 41626 . . . . . . 7 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3029adantr 479 . . . . . 6 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3130eleq2d 2672 . . . . 5 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
323elrab 3330 . . . . 5 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
3331, 32syl6bb 274 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
3418, 28, 33mpbir2and 958 . . 3 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( assIntOp ‘𝑀))
3534ex 448 . 2 (𝑀𝑉 → (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → ∈ ( assIntOp ‘𝑀)))
3614, 35impbid 200 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 194  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172   class class class wbr 4577   × cxp 5026  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7721   assLaw casslaw 41605   clIntOp cclintop 41618   assIntOp cassintop 41619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-asslaw 41609  df-intop 41620  df-clintop 41621  df-assintop 41622
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator