Theorem isablo 30494

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 5927	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2		isabl.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
3	1, 2	eqtr4di 2787	. . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4		raleq 3306	. . . . 5 ⊢ (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
5	4	raleqbi1dv 3321	. . . 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 7419	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8		oveq 7419	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
9	7, 8	eqeq12d 2750	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
10	9	2ralbidv 3208	. . 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 30493	. 2 ⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
13	11, 12	elrab2 3678	1 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ran crn 5666  (class class class)co 7413  GrpOpcgr 30437  AbelOpcablo 30492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-cnv 5673  df-dm 5675  df-rn 5676  df-iota 6494  df-fv 6549  df-ov 7416  df-ablo 30493

This theorem is referenced by:  ablogrpo  30495  ablocom  30496  isabloi  30499