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Theorem ablprop 18411
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.)
Hypotheses
Ref Expression
ablprop.b (Base‘𝐾) = (Base‘𝐿)
ablprop.p (+g𝐾) = (+g𝐿)
Assertion
Ref Expression
ablprop (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Proof of Theorem ablprop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2772 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐾))
2 ablprop.b . . . 4 (Base‘𝐾) = (Base‘𝐿)
32a1i 11 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐿))
4 ablprop.p . . . . 5 (+g𝐾) = (+g𝐿)
54oveqi 6809 . . . 4 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦)
65a1i 11 . . 3 ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
71, 3, 6ablpropd 18410 . 2 (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
87trud 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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