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Theorem ablprop 19036
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 2739 . . 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 7183 . . . 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 19035 . 2 (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
87mptru 1549 1 (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wtru 1543  wcel 2114  cfv 6339  (class class class)co 7170  Basecbs 16586  +gcplusg 16668  Abelcabl 19025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-0g 16818  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-grp 18222  df-cmn 19026  df-abl 19027
This theorem is referenced by:  zlmlmod  20343  dvaabl  38681  cznabel  45066
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