Theorem isablo 30482

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 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑋,𝑦

Proof of Theorem isablo

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rneq 5903	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2		isabl.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
3	1, 2	eqtr4di 2783	. . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4		raleq 3298	. . . . 5 ⊢ (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
5	4	raleqbi1dv 3313	. . . 4 ⊢ (ran 𝑔 = 𝑋 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
6	3, 5	syl 17	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
7		oveq 7396	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8		oveq 7396	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
9	7, 8	eqeq12d 2746	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
10	9	2ralbidv 3202	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
11	6, 10	bitrd 279	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
12		df-ablo 30481	. 2 ⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
13	11, 12	elrab2 3665	1 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ran crn 5642  (class class class)co 7390  GrpOpcgr 30425  AbelOpcablo 30480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652  df-iota 6467  df-fv 6522  df-ov 7393  df-ablo 30481

This theorem is referenced by:  ablogrpo  30483  ablocom  30484  isabloi  30487