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Theorem isablo 27240
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 5315 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2 isabl.1 . . . . 5 𝑋 = ran 𝐺
31, 2syl6eqr 2678 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4 raleq 3132 . . . . 5 (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
54raleqbi1dv 3140 . . . 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 6611 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8 oveq 6611 . . . . 5 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
97, 8eqeq12d 2641 . . . 4 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
1092ralbidv 2988 . . 3 (𝑔 = 𝐺 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
116, 10bitrd 268 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
12 df-ablo 27239 . 2 AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
1311, 12elrab2 3353 1 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  ran crn 5080  (class class class)co 6605  GrpOpcgr 27183  AbelOpcablo 27238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-cnv 5087  df-dm 5089  df-rn 5090  df-iota 5813  df-fv 5858  df-ov 6608  df-ablo 27239
This theorem is referenced by:  ablogrpo  27241  ablocom  27242  isabloi  27245
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