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Theorem grppropstrg 13592
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 13131 . . . . . 6  |-  Base  Fn  _V
3 elex 2812 . . . . . 6  |-  ( K  e.  V  ->  K  e.  _V )
4 funfvex 5652 . . . . . . 7  |-  ( ( Fun  Base  /\  K  e. 
dom  Base )  ->  ( Base `  K )  e. 
_V )
54funfni 5429 . . . . . 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 2317 . . . 4  |-  ( K  e.  V  ->  B  e.  _V )
8 grppropstr.p . . . . 5  |-  ( +g  `  K )  =  .+
9 plusgslid 13185 . . . . . 6  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
109slotex 13099 . . . . 5  |-  ( K  e.  V  ->  ( +g  `  K )  e. 
_V )
118, 10eqeltrrid 2317 . . . 4  |-  ( K  e.  V  ->  .+  e.  _V )
12 grppropstr.l . . . . 5  |-  L  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
1312grpbaseg 13200 . . . 4  |-  ( ( B  e.  _V  /\  .+  e.  _V )  ->  B  =  ( Base `  L ) )
147, 11, 13syl2anc 411 . . 3  |-  ( K  e.  V  ->  B  =  ( Base `  L
) )
151, 14eqtrid 2274 . . 3  |-  ( K  e.  V  ->  ( Base `  K )  =  ( Base `  L
) )
1614, 15eqtr4d 2265 . 2  |-  ( K  e.  V  ->  B  =  ( Base `  K
) )
1712grpplusgg 13201 . . . . 5  |-  ( ( B  e.  _V  /\  .+  e.  _V )  ->  .+  =  ( +g  `  L ) )
187, 11, 17syl2anc 411 . . . 4  |-  ( K  e.  V  ->  .+  =  ( +g  `  L ) )
198, 18eqtrid 2274 . . 3  |-  ( K  e.  V  ->  ( +g  `  K )  =  ( +g  `  L
) )
2019oveqdr 6041 . 2  |-  ( ( K  e.  V  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y ) )
2116, 14, 20grppropd 13590 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 1395    e. wcel 2200   _Vcvv 2800   {cpr 3668   <.cop 3670    Fn wfn 5319   ` cfv 5324   ndxcnx 13069   Basecbs 13072   +g cplusg 13150   Grpcgrp 13573
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576
This theorem is referenced by:  ring1  14062
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