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Theorem grppropstrg 12827
Description: Generalize a specific 2-element group  L to show that any set  K with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grppropstr.b  |-  ( Base `  K )  =  B
grppropstr.p  |-  ( +g  `  K )  =  .+
grppropstr.l  |-  L  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
Assertion
Ref Expression
grppropstrg  |-  ( K  e.  V  ->  ( K  e.  Grp  <->  L  e.  Grp ) )

Proof of Theorem grppropstrg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grppropstr.b . . . . 5  |-  ( Base `  K )  =  B
2 basfn 12512 . . . . . 6  |-  Base  Fn  _V
3 elex 2748 . . . . . 6  |-  ( K  e.  V  ->  K  e.  _V )
4 funfvex 5531 . . . . . . 7  |-  ( ( Fun  Base  /\  K  e. 
dom  Base )  ->  ( Base `  K )  e. 
_V )
54funfni 5315 . . . . . 6  |-  ( (
Base  Fn  _V  /\  K  e.  _V )  ->  ( Base `  K )  e. 
_V )
62, 3, 5sylancr 414 . . . . 5  |-  ( K  e.  V  ->  ( Base `  K )  e. 
_V )
71, 6eqeltrrid 2265 . . . 4  |-  ( K  e.  V  ->  B  e.  _V )
8 grppropstr.p . . . . 5  |-  ( +g  `  K )  =  .+
9 plusgslid 12563 . . . . . 6  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
109slotex 12481 . . . . 5  |-  ( K  e.  V  ->  ( +g  `  K )  e. 
_V )
118, 10eqeltrrid 2265 . . . 4  |-  ( K  e.  V  ->  .+  e.  _V )
12 grppropstr.l . . . . 5  |-  L  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
1312grpbaseg 12577 . . . 4  |-  ( ( B  e.  _V  /\  .+  e.  _V )  ->  B  =  ( Base `  L ) )
147, 11, 13syl2anc 411 . . 3  |-  ( K  e.  V  ->  B  =  ( Base `  L
) )
151, 14eqtrid 2222 . . 3  |-  ( K  e.  V  ->  ( Base `  K )  =  ( Base `  L
) )
1614, 15eqtr4d 2213 . 2  |-  ( K  e.  V  ->  B  =  ( Base `  K
) )
1712grpplusgg 12578 . . . . 5  |-  ( ( B  e.  _V  /\  .+  e.  _V )  ->  .+  =  ( +g  `  L ) )
187, 11, 17syl2anc 411 . . . 4  |-  ( K  e.  V  ->  .+  =  ( +g  `  L ) )
198, 18eqtrid 2222 . . 3  |-  ( K  e.  V  ->  ( +g  `  K )  =  ( +g  `  L
) )
2019oveqdr 5900 . 2  |-  ( ( K  e.  V  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
2116, 14, 20grppropd 12825 1  |-  ( K  e.  V  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737   {cpr 3593   <.cop 3595    Fn wfn 5210   ` cfv 5215   ndxcnx 12451   Basecbs 12454   +g cplusg 12528   Grpcgrp 12809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812
This theorem is referenced by:  ring1  13167
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