ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isgrpid2 Unicode version

Theorem isgrpid2 12907
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 12906 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( Z  .+  Z )  =  Z  <-> 
.0.  =  Z ) )
54biimpd 144 . . 3  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( Z  .+  Z )  =  Z  ->  .0.  =  Z
) )
65expimpd 363 . 2  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  ->  .0.  =  Z
) )
71, 3grpidcl 12898 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
81, 2, 3grplid 12900 . . . . 5  |-  ( ( G  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
97, 8mpdan 421 . . . 4  |-  ( G  e.  Grp  ->  (  .0.  .+  .0.  )  =  .0.  )
107, 9jca 306 . . 3  |-  ( G  e.  Grp  ->  (  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  ) )
11 eleq1 2240 . . . 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 2192 . . . 4  |-  (  .0.  =  Z  ->  (
(  .0.  .+  .0.  )  =  .0.  <->  ( Z  .+  Z )  =  Z ) )
1511, 14anbi12d 473 . . 3  |-  (  .0.  =  Z  ->  (
(  .0.  e.  B  /\  (  .0.  .+  .0.  )  =  .0.  )  <->  ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) ) )
1610, 15syl5ibcom 155 . 2  |-  ( G  e.  Grp  ->  (  .0.  =  Z  ->  ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) ) )
176, 16impbid 129 1  |-  ( G  e.  Grp  ->  (
( Z  e.  B  /\  ( Z  .+  Z
)  =  Z )  <-> 
.0.  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530   0gc0g 12699   Grpcgrp 12871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8918  df-2 8976  df-ndx 12459  df-slot 12460  df-base 12462  df-plusg 12543  df-0g 12701  df-mgm 12769  df-sgrp 12802  df-mnd 12812  df-grp 12874
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator