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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 2610 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
2 | 1 | isablo 26784 | . 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺(𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
3 | 2 | simplbi 475 | 1 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ran crn 5039 (class class class)co 6549 GrpOpcgr 26727 AbelOpcablo 26782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 df-iota 5768 df-fv 5812 df-ov 6552 df-ablo 26783 |
This theorem is referenced by: ablo32 26787 ablo4 26788 ablomuldiv 26790 ablodivdiv 26791 ablodivdiv4 26792 ablonnncan 26794 ablonncan 26795 ablonnncan1 26796 vcgrp 26809 isvcOLD 26818 isvciOLD 26819 cnidOLD 26821 nvgrp 26856 cnnv 26916 cnnvba 26918 cncph 27058 hilid 27402 hhnv 27406 hhba 27408 hhph 27419 hhssabloilem 27502 hhssnv 27505 ablo4pnp 32849 rngogrpo 32879 iscringd 32967 |
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