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Theorem grppropd 15869
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 15759 . . 3  |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
51, 2, 3grpidpropd 15760 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
65adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
73, 6eqeq12d 2489 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
87anass1rs 805 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
98rexbidva 2970 . . . . 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 2900 . . . 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 3065 . . . . 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 3072 . . . 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 3065 . . . . 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 3072 . . . 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 2467 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2467 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
19 eqid 2467 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2017, 18, 19isgrp 15862 . 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 2467 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2467 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
23 eqid 2467 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2421, 22, 23isgrp 15862 . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Mndcmnd 15722   Grpcgrp 15723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-0g 14693  df-mnd 15728  df-grp 15858
This theorem is referenced by:  grpprop  15870  ghmpropd  16099  oppggrpb  16188  ablpropd  16604  rngpropd  17017  lmodprop2d  17355  sralmod  17616  nmpropd2  20850  ngppropd  20886  tngngp2  20901  zhmnrg  27584
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