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Theorem grppropd 14813
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 14711 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
51, 2, 3grpidpropd 14712 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
65adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
73, 6eqeq12d 2449 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
87anass1rs 783 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
98rexbidva 2714 . . . . 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 2713 . . . 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 2903 . . . . 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 2908 . . . 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 2903 . . . . 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 2908 . . . 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 275 . . 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 692 . 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 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2435 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
19 eqid 2435 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2017, 18, 19isgrp 14806 . 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 2435 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2435 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
23 eqid 2435 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2421, 22, 23isgrp 14806 . 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 280 1  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519   0gc0g 13713   Mndcmnd 14674   Grpcgrp 14675
This theorem is referenced by:  grpprop  14814  ghmpropd  15033  oppggrpb  15144  ablpropd  15412  rngpropd  15685  lmodprop2d  15996  sralmod  16248  nmpropd2  18632  ngppropd  18668  tngngp2  18683  zhmnrg  24341
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-0g 13717  df-mnd 14680  df-grp 14802
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