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Mirrors > Home > MPE Home > Th. List > ablogrpo | Structured version Visualization version GIF version |
Description: An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ablogrpo | ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
2 | 1 | isablo 28250 | . 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
3 | 2 | simplbi 498 | 1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ran crn 5549 (class class class)co 7145 GrpOpcgr 28193 AbelOpcablo 28248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 df-iota 6307 df-fv 6356 df-ov 7148 df-ablo 28249 |
This theorem is referenced by: ablo32 28253 ablo4 28254 ablomuldiv 28256 ablodivdiv 28257 ablodivdiv4 28258 ablonncan 28260 ablonnncan1 28261 vcgrp 28274 isvcOLD 28283 isvciOLD 28284 cnidOLD 28286 nvgrp 28321 cnnv 28381 cnnvba 28383 cncph 28523 hilid 28865 hhnv 28869 hhba 28871 hhph 28882 hhssabloilem 28965 hhssnv 28968 ablo4pnp 35039 rngogrpo 35069 iscringd 35157 |
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