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Theorem grpprop 14550
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 2317 . . 3  |-  (  T. 
->  ( Base `  K
)  =  ( Base `  K ) )
2 grpprop.b . . . 4  |-  ( Base `  K )  =  (
Base `  L )
32a1i 10 . . 3  |-  (  T. 
->  ( Base `  K
)  =  ( Base `  L ) )
4 grpprop.p . . . . 5  |-  ( +g  `  K )  =  ( +g  `  L )
54oveqi 5913 . . . 4  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y )
65a1i 10 . . 3  |-  ( (  T.  /\  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
71, 3, 6grppropd 14549 . 2  |-  (  T. 
->  ( K  e.  Grp  <->  L  e.  Grp ) )
87trud 1314 1  |-  ( K  e.  Grp  <->  L  e.  Grp )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1633    e. wcel 1701   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   Grpcgrp 14411
This theorem is referenced by:  grppropstr  14551  grpss  14552  opprrng  15462  opprsubg  15467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-0g 13453  df-mnd 14416  df-grp 14538
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