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Theorem isass 32618
Description: The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
Hypothesis
Ref Expression
isass.1 𝑋 = dom dom 𝐺
Assertion
Ref Expression
isass (𝐺𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
Distinct variable groups:   𝑥,𝐺,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem isass
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 dmeq 5233 . . . . . . . . . 10 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
21dmeqd 5235 . . . . . . . . 9 (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
32eleq2d 2672 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥 ∈ dom dom 𝑔𝑥 ∈ dom dom 𝐺))
42eleq2d 2672 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝐺))
52eleq2d 2672 . . . . . . . 8 (𝑔 = 𝐺 → (𝑧 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝐺))
63, 4, 53anbi123d 1390 . . . . . . 7 (𝑔 = 𝐺 → ((𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔) ↔ (𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺)))
7 oveq 6533 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
87oveq1d 6542 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝑔𝑧))
9 oveq 6533 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝐺𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
108, 9eqtrd 2643 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
11 oveq 6533 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
1211oveq2d 6543 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝑔(𝑦𝐺𝑧)))
13 oveq 6533 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝐺𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
1412, 13eqtrd 2643 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
1510, 14eqeq12d 2624 . . . . . . 7 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
166, 15imbi12d 332 . . . . . 6 (𝑔 = 𝐺 → (((𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))) ↔ ((𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
1716albidv 1835 . . . . 5 (𝑔 = 𝐺 → (∀𝑧((𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))) ↔ ∀𝑧((𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
18172albidv 1837 . . . 4 (𝑔 = 𝐺 → (∀𝑥𝑦𝑧((𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))) ↔ ∀𝑥𝑦𝑧((𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
19 r3al 2923 . . . 4 (∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥𝑦𝑧((𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))))
20 r3al 2923 . . . 4 (∀𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑦𝑧((𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2118, 19, 203bitr4g 301 . . 3 (𝑔 = 𝐺 → (∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
22 isass.1 . . . . . 6 𝑋 = dom dom 𝐺
2322eqcomi 2618 . . . . 5 dom dom 𝐺 = 𝑋
2423a1i 11 . . . 4 (𝑔 = 𝐺 → dom dom 𝐺 = 𝑋)
2524raleqdv 3120 . . . 4 (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2624, 25raleqbidv 3128 . . 3 (𝑔 = 𝐺 → (∀𝑥 ∈ dom dom 𝐺𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2724raleqdv 3120 . . . 4 (𝑔 = 𝐺 → (∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
28272ralbidv 2971 . . 3 (𝑔 = 𝐺 → (∀𝑥𝑋𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2921, 26, 283bitrd 292 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
30 df-ass 32615 . 2 Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
3129, 30elab2g 3321 1 (𝐺𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030  wal 1472   = wceq 1474  wcel 1976  wral 2895  dom cdm 5028  (class class class)co 6527  Asscass 32614
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-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
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-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-dm 5038  df-iota 5754  df-fv 5798  df-ov 6530  df-ass 32615
This theorem is referenced by:  issmgrpOLD  32635
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