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Theorem isgrpid2 15882
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 15881 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( Z  .+  Z )  =  Z  <-> 
.0.  =  Z ) )
54biimpd 207 . . 3  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( Z  .+  Z )  =  Z  ->  .0.  =  Z
) )
65expimpd 603 . 2  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  ->  .0.  =  Z
) )
71, 3grpidcl 15874 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
81, 2, 3grplid 15876 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
97, 8mpdan 668 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  .+  .0.  )  =  .0.  )
107, 9jca 532 . . 3  |-  ( G  e.  Grp  ->  (  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  ) )
11 eleq1 2534 . . . 4  |-  (  .0.  =  Z  ->  (  .0.  e.  B  <->  Z  e.  B ) )
12 id 22 . . . . . 6  |-  (  .0.  =  Z  ->  .0.  =  Z )
1312, 12oveq12d 6295 . . . . 5  |-  (  .0.  =  Z  ->  (  .0.  .+  .0.  )  =  ( Z  .+  Z
) )
1413, 12eqeq12d 2484 . . . 4  |-  (  .0.  =  Z  ->  (
(  .0.  .+  .0.  )  =  .0.  <->  ( Z  .+  Z )  =  Z ) )
1511, 14anbi12d 710 . . 3  |-  (  .0.  =  Z  ->  (
(  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  )  <->  ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) ) )
1610, 15syl5ibcom 220 . 2  |-  ( G  e.  Grp  ->  (  .0.  =  Z  ->  ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) ) )
176, 16impbid 191 1  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  <-> 
.0.  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   0gc0g 14686   Grpcgrp 15718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-riota 6238  df-ov 6280  df-0g 14688  df-mnd 15723  df-grp 15853
This theorem is referenced by:  drngid2  17190  dchr1  23255  erngdvlem4  35664  erngdvlem4-rN  35672
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