MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpss Structured version   Unicode version

Theorem grpss 15881
Description: Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring  R is a group before we know that it is also a ring. (Theorem rnggrp 17005, on the other hand, requires that we know in advance that  R is a ring.) (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpss.g  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
grpss.r  |-  R  e. 
_V
grpss.s  |-  G  C_  R
grpss.f  |-  Fun  R
Assertion
Ref Expression
grpss  |-  ( G  e.  Grp  <->  R  e.  Grp )

Proof of Theorem grpss
StepHypRef Expression
1 grpss.r . . . 4  |-  R  e. 
_V
2 grpss.f . . . 4  |-  Fun  R
3 grpss.s . . . 4  |-  G  C_  R
4 baseid 14536 . . . 4  |-  Base  = Slot  ( Base `  ndx )
5 opex 4711 . . . . . 6  |-  <. ( Base `  ndx ) ,  B >.  e.  _V
65prid1 4135 . . . . 5  |-  <. ( Base `  ndx ) ,  B >.  e.  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. }
7 grpss.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
86, 7eleqtrri 2554 . . . 4  |-  <. ( Base `  ndx ) ,  B >.  e.  G
91, 2, 3, 4, 8strss 14527 . . 3  |-  ( Base `  R )  =  (
Base `  G )
10 plusgid 14590 . . . 4  |-  +g  = Slot  ( +g  `  ndx )
11 opex 4711 . . . . . 6  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  _V
1211prid2 4136 . . . . 5  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. }
1312, 7eleqtrri 2554 . . . 4  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  G
141, 2, 3, 10, 13strss 14527 . . 3  |-  ( +g  `  R )  =  ( +g  `  G )
159, 14grpprop 15879 . 2  |-  ( R  e.  Grp  <->  G  e.  Grp )
1615bicomi 202 1  |-  ( G  e.  Grp  <->  R  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   {cpr 4029   <.cop 4033   Fun wfun 5582   ` cfv 5588   ndxcnx 14487   Basecbs 14490   +g cplusg 14555   Grpcgrp 15727
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  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rrecex 9564  ax-cnre 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-om 6685  df-recs 7042  df-rdg 7076  df-nn 10537  df-2 10594  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-0g 14697  df-mnd 15732  df-grp 15867
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator