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Theorem isgrpid2 14518
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 14517 . . . 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 14510 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
81, 2, 3grplid 14512 . . . . 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 2343 . . . 4  |-  (  .0.  =  Z  ->  (  .0.  e.  B  <->  Z  e.  B ) )
12 id 19 . . . . . 6  |-  (  .0.  =  Z  ->  .0.  =  Z )
1312, 12oveq12d 5876 . . . . 5  |-  (  .0.  =  Z  ->  (  .0.  .+  .0.  )  =  ( Z  .+  Z
) )
1413, 12eqeq12d 2297 . . . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362
This theorem is referenced by:  drngid2  15528  dchr1  20496  erngdvlem4  31180  erngdvlem4-rN  31188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489
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