Theorem isablo 30879

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 5928	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
2		isabl.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
3	1, 2	eqtr4di 2816	. . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
4		raleq 3320	. . . . 5 ⊢ (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
5	4	raleqbi1dv 3333	. . . 4 ⊢ (ran 𝑔 = 𝑋 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
6	3, 5	syl 18	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥)))
7		oveq 7418	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8		oveq 7418	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
9	7, 8	eqeq12d 2779	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
10	9	2ralbidv 3229	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
11	6, 10	bitrd 282	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
12		df-ablo 30878	. 2 ⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
13	11, 12	elrab2 3655	1 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ran crn 5664  (class class class)co 7412  GrpOpcgr 30822  AbelOpcablo 30877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-cnv 5671  df-dm 5673  df-rn 5674  df-iota 6494  df-fv 6546  df-ov 7415  df-ablo 30878

This theorem is referenced by:  ablogrpo  30880  ablocom  30881  isabloi  30884