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Theorem grppropd 14516
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grppropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
grppropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
grppropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
Assertion
Ref Expression
grppropd  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem grppropd
StepHypRef Expression
1 grppropd.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 grppropd.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
3 grppropd.3 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
41, 2, 3mndpropd 14414 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
51, 2, 3grpidpropd 14415 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
65adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
73, 6eqeq12d 2310 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
87anass1rs 782 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
98rexbidva 2573 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  ( E. x  e.  B  ( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  E. x  e.  B  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
109ralbidva 2572 . . . 4  |-  ( ph  ->  ( A. y  e.  B  E. x  e.  B  ( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  A. y  e.  B  E. x  e.  B  ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
111rexeqdv 2756 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  E. x  e.  ( Base `  K ) ( x ( +g  `  K
) y )  =  ( 0g `  K
) ) )
121, 11raleqbidv 2761 . . . 4  |-  ( ph  ->  ( A. y  e.  B  E. x  e.  B  ( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  A. y  e.  ( Base `  K ) E. x  e.  ( Base `  K ) ( x ( +g  `  K
) y )  =  ( 0g `  K
) ) )
132rexeqdv 2756 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( +g  `  L ) y )  =  ( 0g `  L )  <->  E. x  e.  ( Base `  L ) ( x ( +g  `  L
) y )  =  ( 0g `  L
) ) )
142, 13raleqbidv 2761 . . . 4  |-  ( ph  ->  ( A. y  e.  B  E. x  e.  B  ( x ( +g  `  L ) y )  =  ( 0g `  L )  <->  A. y  e.  ( Base `  L ) E. x  e.  ( Base `  L ) ( x ( +g  `  L
) y )  =  ( 0g `  L
) ) )
1510, 12, 143bitr3d 274 . . 3  |-  ( ph  ->  ( A. y  e.  ( Base `  K
) E. x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  A. y  e.  ( Base `  L ) E. x  e.  ( Base `  L ) ( x ( +g  `  L
) y )  =  ( 0g `  L
) ) )
164, 15anbi12d 691 . 2  |-  ( ph  ->  ( ( K  e. 
Mnd  /\  A. y  e.  ( Base `  K
) E. x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) )  <->  ( L  e. 
Mnd  /\  A. y  e.  ( Base `  L
) E. x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) ) )
17 eqid 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2296 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
19 eqid 2296 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2017, 18, 19isgrp 14509 . 2  |-  ( K  e.  Grp  <->  ( K  e.  Mnd  /\  A. y  e.  ( Base `  K
) E. x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
21 eqid 2296 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2296 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
23 eqid 2296 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2421, 22, 23isgrp 14509 . 2  |-  ( L  e.  Grp  <->  ( L  e.  Mnd  /\  A. y  e.  ( Base `  L
) E. x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
2516, 20, 243bitr4g 279 1  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378
This theorem is referenced by:  grpprop  14517  ghmpropd  14736  oppggrpb  14847  ablpropd  15115  rngpropd  15388  lmodprop2d  15703  sralmod  15955  nmpropd2  18133  ngppropd  18169  tngngp2  18184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-0g 13420  df-mnd 14383  df-grp 14505
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