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Theorem mndprop 14723
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
Hypotheses
Ref Expression
mndprop.b  |-  ( Base `  K )  =  (
Base `  L )
mndprop.p  |-  ( +g  `  K )  =  ( +g  `  L )
Assertion
Ref Expression
mndprop  |-  ( K  e.  Mnd  <->  L  e.  Mnd )

Proof of Theorem mndprop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2437 . . 3  |-  (  T. 
->  ( Base `  K
)  =  ( Base `  K ) )
2 mndprop.b . . . 4  |-  ( Base `  K )  =  (
Base `  L )
32a1i 11 . . 3  |-  (  T. 
->  ( Base `  K
)  =  ( Base `  L ) )
4 mndprop.p . . . . 5  |-  ( +g  `  K )  =  ( +g  `  L )
54oveqi 6094 . . . 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, 6mndpropd 14721 . 2  |-  (  T. 
->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
87trud 1332 1  |-  ( K  e.  Mnd  <->  L  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Mndcmnd 14684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338  ax-pow 4377
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-mnd 14690
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