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Theorem grpprop 18983
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpprop.b (Base‘𝐾) = (Base‘𝐿)
grpprop.p (+g𝐾) = (+g𝐿)
Assertion
Ref Expression
grpprop (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Proof of Theorem grpprop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2736 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐾))
2 grpprop.b . . . 4 (Base‘𝐾) = (Base‘𝐿)
32a1i 11 . . 3 (⊤ → (Base‘𝐾) = (Base‘𝐿))
4 grpprop.p . . . . 5 (+g𝐾) = (+g𝐿)
54oveqi 7444 . . . 4 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦)
65a1i 11 . . 3 ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
71, 3, 6grppropd 18982 . 2 (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
87mptru 1544 1 (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wtru 1538  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967
This theorem is referenced by:  grppropstr  18984  grpss  18985  opprsubg  20369  rmodislmod  20945  rmodislmodOLD  20946  opprgrpb  42499  lmod1  48338
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