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Theorem isgrpid2 16302
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 16301 . . . 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 601 . 2  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  ->  .0.  =  Z
) )
71, 3grpidcl 16294 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
81, 2, 3grplid 16296 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
97, 8mpdan 666 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  .+  .0.  )  =  .0.  )
107, 9jca 530 . . 3  |-  ( G  e.  Grp  ->  (  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  ) )
11 eleq1 2474 . . . 4  |-  (  .0.  =  Z  ->  (  .0.  e.  B  <->  Z  e.  B ) )
12 id 22 . . . . . 6  |-  (  .0.  =  Z  ->  .0.  =  Z )
1312, 12oveq12d 6252 . . . . 5  |-  (  .0.  =  Z  ->  (  .0.  .+  .0.  )  =  ( Z  .+  Z
) )
1413, 12eqeq12d 2424 . . . 4  |-  (  .0.  =  Z  ->  (
(  .0.  .+  .0.  )  =  .0.  <->  ( Z  .+  Z )  =  Z ) )
1511, 14anbi12d 709 . . 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 367    = wceq 1405    e. wcel 1842   ` cfv 5525  (class class class)co 6234   Basecbs 14733   +g cplusg 14801   0gc0g 14946   Grpcgrp 16269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fv 5533  df-riota 6196  df-ov 6237  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-grp 16273
This theorem is referenced by:  drngid2  17624  dchr1  23805  erngdvlem4  33991  erngdvlem4-rN  33999
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