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| Description: Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| isabli.1 | ⊢ 𝐺 ∈ GrpOp | 
| isabli.2 | ⊢ dom 𝐺 = (𝑋 × 𝑋) | 
| isabli.3 | ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) | 
| Ref | Expression | 
|---|---|
| isabloi | ⊢ 𝐺 ∈ AbelOp | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isabli.1 | . 2 ⊢ 𝐺 ∈ GrpOp | |
| 2 | isabli.3 | . . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) | |
| 3 | 2 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | 
| 4 | isabli.2 | . . . 4 ⊢ dom 𝐺 = (𝑋 × 𝑋) | |
| 5 | 1, 4 | grporn 30540 | . . 3 ⊢ 𝑋 = ran 𝐺 | 
| 6 | 5 | isablo 30565 | . 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) | 
| 7 | 1, 3, 6 | mpbir2an 711 | 1 ⊢ 𝐺 ∈ AbelOp | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 × cxp 5683 dom cdm 5685 (class class class)co 7431 GrpOpcgr 30508 AbelOpcablo 30563 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-grpo 30512 df-ablo 30564 | 
| This theorem is referenced by: cnaddabloOLD 30600 hilablo 31179 hhssabloi 31281 | 
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