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Theorem grpprop 17485
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 2652 . . 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 6703 . . . 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 17484 . 2 (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
87trud 1533 1 (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wtru 1524  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472
This theorem is referenced by:  grppropstr  17486  grpss  17487  opprring  18677  opprsubg  18682  rmodislmod  18979  lmod1  42606
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