MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgrpid2 Unicode version

Theorem isgrpid2 14534
Description: Properties showing that an element  Z is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinveu.b  |-  B  =  ( Base `  G
)
grpinveu.p  |-  .+  =  ( +g  `  G )
grpinveu.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
isgrpid2  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  <-> 
.0.  =  Z ) )

Proof of Theorem isgrpid2
StepHypRef Expression
1 grpinveu.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpinveu.p . . . . 5  |-  .+  =  ( +g  `  G )
3 grpinveu.o . . . . 5  |-  .0.  =  ( 0g `  G )
41, 2, 3grpid 14533 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( Z  .+  Z )  =  Z  <-> 
.0.  =  Z ) )
54biimpd 198 . . 3  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( Z  .+  Z )  =  Z  ->  .0.  =  Z
) )
65expimpd 586 . 2  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  ->  .0.  =  Z
) )
71, 3grpidcl 14526 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
81, 2, 3grplid 14528 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
97, 8mpdan 649 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  .+  .0.  )  =  .0.  )
107, 9jca 518 . . 3  |-  ( G  e.  Grp  ->  (  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  ) )
11 eleq1 2356 . . . 4  |-  (  .0.  =  Z  ->  (  .0.  e.  B  <->  Z  e.  B ) )
12 id 19 . . . . . 6  |-  (  .0.  =  Z  ->  .0.  =  Z )
1312, 12oveq12d 5892 . . . . 5  |-  (  .0.  =  Z  ->  (  .0.  .+  .0.  )  =  ( Z  .+  Z
) )
1413, 12eqeq12d 2310 . . . 4  |-  (  .0.  =  Z  ->  (
(  .0.  .+  .0.  )  =  .0.  <->  ( Z  .+  Z )  =  Z ) )
1511, 14anbi12d 691 . . 3  |-  (  .0.  =  Z  ->  (
(  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  )  <->  ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) ) )
1610, 15syl5ibcom 211 . 2  |-  ( G  e.  Grp  ->  (  .0.  =  Z  ->  ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) ) )
176, 16impbid 183 1  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  <-> 
.0.  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378
This theorem is referenced by:  drngid2  15544  dchr1  20512  erngdvlem4  31802  erngdvlem4-rN  31810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505
  Copyright terms: Public domain W3C validator