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 37833
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 5917 . . . . . . . . . 10 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
21dmeqd 5919 . . . . . . . . 9 (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
32eleq2d 2825 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥 ∈ dom dom 𝑔𝑥 ∈ dom dom 𝐺))
42eleq2d 2825 . . . . . . . 8 (𝑔 = 𝐺 → (𝑦 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝐺))
52eleq2d 2825 . . . . . . . 8 (𝑔 = 𝐺 → (𝑧 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝐺))
63, 4, 53anbi123d 1435 . . . . . . 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 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
87oveq1d 7446 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝑔𝑧))
9 oveq 7437 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝐺𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
108, 9eqtrd 2775 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
11 oveq 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
1211oveq2d 7447 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝑔(𝑦𝐺𝑧)))
13 oveq 7437 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝐺𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
1412, 13eqtrd 2775 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
1510, 14eqeq12d 2751 . . . . . . 7 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
166, 15imbi12d 344 . . . . . 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 1918 . . . . 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 1921 . . . 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 3195 . . . 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 3195 . . . 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 314 . . 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 2744 . . . . 5 dom dom 𝐺 = 𝑋
2423a1i 11 . . . 4 (𝑔 = 𝐺 → dom dom 𝐺 = 𝑋)
2524raleqdv 3324 . . . 4 (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝐺𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2624, 25raleqbidv 3344 . . 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 3324 . . . 4 (𝑔 = 𝐺 → (∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
28272ralbidv 3219 . . 3 (𝑔 = 𝐺 → (∀𝑥𝑋𝑦𝑋𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
2921, 26, 283bitrd 305 . 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 37830 . 2 Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
3129, 30elab2g 3683 1 (𝐺𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  dom cdm 5689  (class class class)co 7431  Asscass 37829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-ass 37830
This theorem is referenced by:  issmgrpOLD  37850
  Copyright terms: Public domain W3C validator