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Theorem isablo 30369
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 5938 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2 isabl.1 . . . . 5 𝑋 = ran 𝐺
31, 2eqtr4di 2786 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4 raleq 3319 . . . . 5 (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
54raleqbi1dv 3330 . . . 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 7426 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8 oveq 7426 . . . . 5 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
97, 8eqeq12d 2744 . . . 4 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
1092ralbidv 3215 . . 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 30368 . 2 AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
1311, 12elrab2 3685 1 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  ran crn 5679  (class class class)co 7420  GrpOpcgr 30312  AbelOpcablo 30367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5686  df-dm 5688  df-rn 5689  df-iota 6500  df-fv 6556  df-ov 7423  df-ablo 30368
This theorem is referenced by:  ablogrpo  30370  ablocom  30371  isabloi  30374
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