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Theorem isablo 30575
Description: The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
isabl.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isablo (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦

Proof of Theorem isablo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 rneq 5950 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2 isabl.1 . . . . 5 𝑋 = ran 𝐺
31, 2eqtr4di 2793 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4 raleq 3321 . . . . 5 (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
54raleqbi1dv 3336 . . . 4 (ran 𝑔 = 𝑋 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
63, 5syl 17 . . 3 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
7 oveq 7437 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8 oveq 7437 . . . . 5 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
97, 8eqeq12d 2751 . . . 4 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
1092ralbidv 3219 . . 3 (𝑔 = 𝐺 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
116, 10bitrd 279 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
12 df-ablo 30574 . 2 AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
1311, 12elrab2 3698 1 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  ran crn 5690  (class class class)co 7431  GrpOpcgr 30518  AbelOpcablo 30573
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-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571  df-ov 7434  df-ablo 30574
This theorem is referenced by:  ablogrpo  30576  ablocom  30577  isabloi  30580
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