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Theorem ablprop 18910
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 2820 . . 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 7161 . . . 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 18909 . 2 (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
87mptru 1538 1 (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1531  wtru 1532  wcel 2108  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  Abelcabl 18899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-cmn 18900  df-abl 18901
This theorem is referenced by:  zlmlmod  20662  dvaabl  38152  cznabel  44216
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