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Theorem ablprop 18125
 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 2622 . . 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 6617 . . . 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 18124 . 2 (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
87trud 1490 1 (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480  ⊤wtru 1481   ∈ wcel 1987  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Abelcabl 18115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-cmn 18116  df-abl 18117 This theorem is referenced by:  zlmlmod  19790  dvaabl  35793  cznabel  41242
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