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Theorem grppropd 14500
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 14398 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
51, 2, 3grpidpropd 14399 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
65adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
73, 6eqeq12d 2297 . . . . . . 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 2560 . . . . 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 2559 . . . 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 2743 . . . . 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 2748 . . . 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 2743 . . . . 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 2748 . . . 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 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2283 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
19 eqid 2283 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2017, 18, 19isgrp 14493 . 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 2283 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2283 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
23 eqid 2283 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2421, 22, 23isgrp 14493 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362
This theorem is referenced by:  grpprop  14501  ghmpropd  14720  oppggrpb  14831  ablpropd  15099  rngpropd  15372  lmodprop2d  15687  sralmod  15939  nmpropd2  18117  ngppropd  18153  tngngp2  18168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-0g 13404  df-mnd 14367  df-grp 14489
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