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Theorem mndprop 15568
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 2455 . . 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 6214 . . . 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 15566 . 2  |-  ( T. 
->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
87trud 1379 1  |-  ( K  e.  Mnd  <->  L  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   T. wtru 1371    e. wcel 1758   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   Mndcmnd 15529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4530  ax-pow 4579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-ov 6204  df-mnd 15535
This theorem is referenced by:  rng1  31158
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