Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isass Structured version   Visualization version   GIF version

Theorem isass 33775
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 5356 . . . . . . . . . 10 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
21dmeqd 5358 . . . . . . . . 9 (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
32eleq2d 2716 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥 ∈ dom dom 𝑔𝑥 ∈ dom dom 𝐺))
42eleq2d 2716 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝐺))
52eleq2d 2716 . . . . . . . 8 (𝑔 = 𝐺 → (𝑧 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝐺))
63, 4, 53anbi123d 1439 . . . . . . 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 6696 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
87oveq1d 6705 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝑔𝑧))
9 oveq 6696 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝐺𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
108, 9eqtrd 2685 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
11 oveq 6696 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
1211oveq2d 6706 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝑔(𝑦𝐺𝑧)))
13 oveq 6696 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝐺𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
1412, 13eqtrd 2685 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
1510, 14eqeq12d 2666 . . . . . . 7 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
166, 15imbi12d 333 . . . . . 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 1889 . . . . 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 1891 . . . 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 2969 . . . 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 2969 . . . 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 303 . . 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 2660 . . . . 5 dom dom 𝐺 = 𝑋
2423a1i 11 . . . 4 (𝑔 = 𝐺 → dom dom 𝐺 = 𝑋)
2524raleqdv 3174 . . . 4 (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2624, 25raleqbidv 3182 . . 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 3174 . . . 4 (𝑔 = 𝐺 → (∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
28272ralbidv 3018 . . 3 (𝑔 = 𝐺 → (∀𝑥𝑋𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2921, 26, 283bitrd 294 . 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 33772 . 2 Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
3129, 30elab2g 3385 1 (𝐺𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wral 2941  dom cdm 5143  (class class class)co 6690  Asscass 33771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693  df-ass 33772
This theorem is referenced by:  issmgrpOLD  33792
  Copyright terms: Public domain W3C validator