Theorem isabloi 29021

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 3191	. 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
4		isabli.2	. . . 4 ⊢ dom 𝐺 = (𝑋 × 𝑋)
5	1, 4	grporn 28991	. . 3 ⊢ 𝑋 = ran 𝐺
6	5	isablo 29016	. 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
7	1, 3, 6	mpbir2an 708	1 ⊢ 𝐺 ∈ AbelOp

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062   × cxp 5603  dom cdm 5605  (class class class)co 7313  GrpOpcgr 28959  AbelOpcablo 29014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fo 6469  df-fv 6471  df-ov 7316  df-grpo 28963  df-ablo 29015

This theorem is referenced by:  cnaddabloOLD  29051  hilablo  29630  hhssabloi  29732