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Theorem grpprop 17362
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 2622 . . 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 6620 . . . 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 17361 . 2 (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
87trud 1490 1 (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  cfv 5849  (class class class)co 6607  Basecbs 15784  +gcplusg 15865  Grpcgrp 17346
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 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
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 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-ov 6610  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349
This theorem is referenced by:  grppropstr  17363  grpss  17364  opprring  18555  opprsubg  18560  lmod1  41585
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