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Theorem grpidd2 14519
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14507. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
grpidd2.b  |-  ( ph  ->  B  =  ( Base `  G ) )
grpidd2.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
grpidd2.z  |-  ( ph  ->  .0.  e.  B )
grpidd2.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
grpidd2.j  |-  ( ph  ->  G  e.  Grp )
Assertion
Ref Expression
grpidd2  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Distinct variable groups:    x, B    x, 
.+    ph, x    x,  .0.
Allowed substitution hint:    G( x)

Proof of Theorem grpidd2
StepHypRef Expression
1 grpidd2.p . . . . 5  |-  ( ph  ->  .+  =  ( +g  `  G ) )
21oveqd 5875 . . . 4  |-  ( ph  ->  (  .0.  .+  .0.  )  =  (  .0.  ( +g  `  G )  .0.  ) )
3 grpidd2.z . . . . 5  |-  ( ph  ->  .0.  e.  B )
4 grpidd2.i . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
54ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. x  e.  B  (  .0.  .+  x )  =  x )
6 oveq2 5866 . . . . . . 7  |-  ( x  =  .0.  ->  (  .0.  .+  x )  =  (  .0.  .+  .0.  ) )
7 id 19 . . . . . . 7  |-  ( x  =  .0.  ->  x  =  .0.  )
86, 7eqeq12d 2297 . . . . . 6  |-  ( x  =  .0.  ->  (
(  .0.  .+  x
)  =  x  <->  (  .0.  .+  .0.  )  =  .0.  ) )
98rspcv 2880 . . . . 5  |-  (  .0. 
e.  B  ->  ( A. x  e.  B  (  .0.  .+  x )  =  x  ->  (  .0.  .+  .0.  )  =  .0.  ) )
103, 5, 9sylc 56 . . . 4  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
112, 10eqtr3d 2317 . . 3  |-  ( ph  ->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
12 grpidd2.j . . . 4  |-  ( ph  ->  G  e.  Grp )
13 grpidd2.b . . . . 5  |-  ( ph  ->  B  =  ( Base `  G ) )
143, 13eleqtrd 2359 . . . 4  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
15 eqid 2283 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
16 eqid 2283 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
17 eqid 2283 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
1815, 16, 17grpid 14517 . . . 4  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (
(  .0.  ( +g  `  G )  .0.  )  =  .0.  <->  ( 0g `  G )  =  .0.  ) )
1912, 14, 18syl2anc 642 . . 3  |-  ( ph  ->  ( (  .0.  ( +g  `  G )  .0.  )  =  .0.  <->  ( 0g `  G )  =  .0.  ) )
2011, 19mpbid 201 . 2  |-  ( ph  ->  ( 0g `  G
)  =  .0.  )
2120eqcomd 2288 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362
This theorem is referenced by:  imasgrp2  14610
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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