Theorem isabloi 30532

Description: Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isabli.1	⊢ 𝐺 ∈ GrpOp
isabli.2	⊢ dom 𝐺 = (𝑋 × 𝑋)
isabli.3	⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))

Assertion

Ref	Expression
isabloi	⊢ 𝐺 ∈ AbelOp

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

Proof of Theorem isabloi

Step	Hyp	Ref	Expression
1		isabli.1	. 2 ⊢ 𝐺 ∈ GrpOp
2		isabli.3	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
3	2	rgen2 3184	. 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
4		isabli.2	. . . 4 ⊢ dom 𝐺 = (𝑋 × 𝑋)
5	1, 4	grporn 30502	. . 3 ⊢ 𝑋 = ran 𝐺
6	5	isablo 30527	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
7	1, 3, 6	mpbir2an 711	1 ⊢ 𝐺 ∈ AbelOp

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   × cxp 5652  dom cdm 5654  (class class class)co 7405  GrpOpcgr 30470  AbelOpcablo 30525

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-ov 7408  df-grpo 30474  df-ablo 30526

This theorem is referenced by:  cnaddabloOLD  30562  hilablo  31141  hhssabloi  31243