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Theorem isassintop 42447
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 42443 . . . . 5 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
21eleq2d 2830 . . . 4 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3 breq1 4812 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
43elrab 3519 . . . 4 ( ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀))
52, 4syl6bb 278 . . 3 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)))
6 elmapi 8082 . . . . . 6 ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76ad2antrl 719 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
8 isasslaw 42429 . . . . . . . 8 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
98biimpd 220 . . . . . . 7 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ 𝑀𝑉) → ( assLaw 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
109impancom 443 . . . . . 6 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀) → (𝑀𝑉 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1110impcom 396 . . . . 5 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
127, 11jca 507 . . . 4 ((𝑀𝑉 ∧ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀)) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1312ex 401 . . 3 (𝑀𝑉 → (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
145, 13sylbid 231 . 2 (𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
15 isclintop 42444 . . . . . . 7 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
1615biimprcd 241 . . . . . 6 ( :(𝑀 × 𝑀)⟶𝑀 → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1716adantr 472 . . . . 5 (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → (𝑀𝑉 ∈ ( clIntOp ‘𝑀)))
1817impcom 396 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( clIntOp ‘𝑀))
19 sqxpexg 7161 . . . . . . . . 9 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
20 fex 6682 . . . . . . . . 9 (( :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ∈ V)
2119, 20sylan2 586 . . . . . . . 8 (( :(𝑀 × 𝑀)⟶𝑀𝑀𝑉) → ∈ V)
2221ancoms 450 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → ∈ V)
23 simpl 474 . . . . . . 7 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → 𝑀𝑉)
24 isasslaw 42429 . . . . . . . 8 (( ∈ V ∧ 𝑀𝑉) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2524bicomd 214 . . . . . . 7 (( ∈ V ∧ 𝑀𝑉) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2622, 23, 25syl2anc 579 . . . . . 6 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ assLaw 𝑀))
2726biimpd 220 . . . . 5 ((𝑀𝑉 :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → assLaw 𝑀))
2827impr 446 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → assLaw 𝑀)
29 assintopval 42442 . . . . . . 7 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3029adantr 472 . . . . . 6 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3130eleq2d 2830 . . . . 5 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
323elrab 3519 . . . . 5 ( ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀))
3331, 32syl6bb 278 . . . 4 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ( ∈ ( assIntOp ‘𝑀) ↔ ( ∈ ( clIntOp ‘𝑀) ∧ assLaw 𝑀)))
3418, 28, 33mpbir2and 704 . . 3 ((𝑀𝑉 ∧ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))) → ∈ ( assIntOp ‘𝑀))
3534ex 401 . 2 (𝑀𝑉 → (( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) → ∈ ( assIntOp ‘𝑀)))
3614, 35impbid 203 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350   class class class wbr 4809   × cxp 5275  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060   assLaw casslaw 42421   clIntOp cclintop 42434   assIntOp cassintop 42435
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-rep 4930  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-reu 3062  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-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-asslaw 42425  df-intop 42436  df-clintop 42437  df-assintop 42438
This theorem is referenced by: (None)
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