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Theorem grpprop 16393
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 2403 . . 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 6291 . . . 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 16392 . 2  |-  ( T. 
->  ( K  e.  Grp  <->  L  e.  Grp ) )
87trud 1414 1  |-  ( K  e.  Grp  <->  L  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   Grpcgrp 16377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381
This theorem is referenced by:  grppropstr  16394  grpss  16395  opprring  17600  opprsubg  17605  lmod1  38604
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