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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 3168 | . 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) |
4 | isabli.2 | . . . 4 ⊢ dom 𝐺 = (𝑋 × 𝑋) | |
5 | 1, 4 | grporn 28304 | . . 3 ⊢ 𝑋 = ran 𝐺 |
6 | 5 | isablo 28329 | . 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
7 | 1, 3, 6 | mpbir2an 710 | 1 ⊢ 𝐺 ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 × cxp 5517 dom cdm 5519 (class class class)co 7135 GrpOpcgr 28272 AbelOpcablo 28327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-ov 7138 df-grpo 28276 df-ablo 28328 |
This theorem is referenced by: cnaddabloOLD 28364 hilablo 28943 hhssabloi 29045 |
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