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Mirrors > Home > MPE Home > Th. List > ablprop | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.) |
Ref | Expression |
---|---|
ablprop.b | ⊢ (Base‘𝐾) = (Base‘𝐿) |
ablprop.p | ⊢ (+g‘𝐾) = (+g‘𝐿) |
Ref | Expression |
---|---|
ablprop | ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2772 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾)) | |
2 | ablprop.b | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿)) |
4 | ablprop.p | . . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
5 | 4 | oveqi 6809 | . . . 4 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) |
6 | 5 | a1i 11 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
7 | 1, 3, 6 | ablpropd 18410 | . 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)) |
8 | 7 | trud 1641 | 1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ⊤wtru 1632 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 +gcplusg 16149 Abelcabl 18401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5993 df-fun 6032 df-fv 6038 df-ov 6799 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-cmn 18402 df-abl 18403 |
This theorem is referenced by: zlmlmod 20086 dvaabl 36834 cznabel 42479 |
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