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Theorem mndprop 16149
Description: If two structures have the same group components (properties), one is a monoid 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 6283 . . . 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 16148 . 2  |-  ( T. 
->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
87trud 1407 1  |-  ( K  e.  Mnd  <->  L  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398   T. wtru 1399    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   Mndcmnd 16121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568  ax-pow 4615
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-mgm 16074  df-sgrp 16113  df-mnd 16123
This theorem is referenced by:  ring1  17446
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