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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 2622 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
2 | 1 | isablo 27400 | . 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
3 | 2 | simplbi 476 | 1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ran crn 5115 (class class class)co 6650 GrpOpcgr 27343 AbelOpcablo 27398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 df-ov 6653 df-ablo 27399 |
This theorem is referenced by: ablo32 27403 ablo4 27404 ablomuldiv 27406 ablodivdiv 27407 ablodivdiv4 27408 ablonnncan 27410 ablonncan 27411 ablonnncan1 27412 vcgrp 27425 isvcOLD 27434 isvciOLD 27435 cnidOLD 27437 nvgrp 27472 cnnv 27532 cnnvba 27534 cncph 27674 hilid 28018 hhnv 28022 hhba 28024 hhph 28035 hhssabloilem 28118 hhssnv 28121 ablo4pnp 33679 rngogrpo 33709 iscringd 33797 |
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