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Theorem grppropd 16194
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 16072 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
51, 2, 3grpidpropd 16014 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
65adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
73, 6eqeq12d 2479 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
87anass1rs 807 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
98rexbidva 2965 . . . . 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 2893 . . . 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 3061 . . . . 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 3068 . . . 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 3061 . . . . 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 3068 . . . 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 283 . . 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 710 . 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 2457 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2457 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
19 eqid 2457 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2017, 18, 19isgrp 16187 . 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 2457 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2457 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
23 eqid 2457 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2421, 22, 23isgrp 16187 . 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 288 1  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856   Mndcmnd 16045   Grpcgrp 16179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183
This theorem is referenced by:  grpprop  16195  ghmpropd  16430  oppggrpb  16519  ablpropd  16934  ringpropd  17356  lmodprop2d  17698  sralmod  17959  nmpropd2  21240  ngppropd  21276  tngngp2  21291  zhmnrg  28101
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