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Theorem grpprop 15863
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  |-  ( +g  `  K )  =  ( +g  `  L )
Assertion
Ref Expression
grpprop  |-  ( K  e.  Grp  <->  L  e.  Grp )

Proof of Theorem grpprop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2461 . . 3  |-  ( T. 
->  ( Base `  K
)  =  ( Base `  K ) )
2 grpprop.b . . . 4  |-  ( Base `  K )  =  (
Base `  L )
32a1i 11 . . 3  |-  ( T. 
->  ( Base `  K
)  =  ( Base `  L ) )
4 grpprop.p . . . . 5  |-  ( +g  `  K )  =  ( +g  `  L )
54oveqi 6288 . . . 4  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y )
65a1i 11 . . 3  |-  ( ( T.  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
71, 3, 6grppropd 15862 . 2  |-  ( T. 
->  ( K  e.  Grp  <->  L  e.  Grp ) )
87trud 1383 1  |-  ( K  e.  Grp  <->  L  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   T. wtru 1375    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-0g 14686  df-mnd 15721  df-grp 15851
This theorem is referenced by:  grppropstr  15864  grpss  15865  opprrng  17057  opprsubg  17062  lmod1  32049
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