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Theorem isablo 28250
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 5799 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2 isabl.1 . . . . 5 𝑋 = ran 𝐺
31, 2syl6eqr 2871 . . . 4 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4 raleq 3403 . . . . 5 (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
54raleqbi1dv 3401 . . . 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 7151 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8 oveq 7151 . . . . 5 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
97, 8eqeq12d 2834 . . . 4 (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
1092ralbidv 3196 . . 3 (𝑔 = 𝐺 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
116, 10bitrd 280 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
12 df-ablo 28249 . 2 AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
1311, 12elrab2 3680 1 (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  ran crn 5549  (class class class)co 7145  GrpOpcgr 28193  AbelOpcablo 28248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356  df-ov 7148  df-ablo 28249
This theorem is referenced by:  ablogrpo  28251  ablocom  28252  isabloi  28255
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