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Theorem grpidd2 14535
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14523. (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 5891 . . . 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 2639 . . . . 5  |-  ( ph  ->  A. x  e.  B  (  .0.  .+  x )  =  x )
6 oveq2 5882 . . . . . . 7  |-  ( x  =  .0.  ->  (  .0.  .+  x )  =  (  .0.  .+  .0.  ) )
7 id 19 . . . . . . 7  |-  ( x  =  .0.  ->  x  =  .0.  )
86, 7eqeq12d 2310 . . . . . 6  |-  ( x  =  .0.  ->  (
(  .0.  .+  x
)  =  x  <->  (  .0.  .+  .0.  )  =  .0.  ) )
98rspcv 2893 . . . . 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 2330 . . 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 2372 . . . 4  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
15 eqid 2296 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
16 eqid 2296 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
17 eqid 2296 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
1815, 16, 17grpid 14533 . . . 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 2301 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378
This theorem is referenced by:  imasgrp2  14626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505
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