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Mirrors > Home > MPE Home > Th. List > isabloi | Structured version Visualization version GIF version |
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 3197 | . 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) |
4 | isabli.2 | . . . 4 ⊢ dom 𝐺 = (𝑋 × 𝑋) | |
5 | 1, 4 | grporn 30550 | . . 3 ⊢ 𝑋 = ran 𝐺 |
6 | 5 | isablo 30575 | . 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 1537 ∈ wcel 2106 ∀wral 3059 × cxp 5687 dom cdm 5689 (class class class)co 7431 GrpOpcgr 30518 AbelOpcablo 30573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-grpo 30522 df-ablo 30574 |
This theorem is referenced by: cnaddabloOLD 30610 hilablo 31189 hhssabloi 31291 |
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