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Theorem grpidd2 14842
Description: Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14830. (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 6098 . . . 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 2789 . . . . 5  |-  ( ph  ->  A. x  e.  B  (  .0.  .+  x )  =  x )
6 oveq2 6089 . . . . . . 7  |-  ( x  =  .0.  ->  (  .0.  .+  x )  =  (  .0.  .+  .0.  ) )
7 id 20 . . . . . . 7  |-  ( x  =  .0.  ->  x  =  .0.  )
86, 7eqeq12d 2450 . . . . . 6  |-  ( x  =  .0.  ->  (
(  .0.  .+  x
)  =  x  <->  (  .0.  .+  .0.  )  =  .0.  ) )
98rspcv 3048 . . . . 5  |-  (  .0. 
e.  B  ->  ( A. x  e.  B  (  .0.  .+  x )  =  x  ->  (  .0.  .+  .0.  )  =  .0.  ) )
103, 5, 9sylc 58 . . . 4  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
112, 10eqtr3d 2470 . . 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 2512 . . . 4  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
15 eqid 2436 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
16 eqid 2436 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
17 eqid 2436 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
1815, 16, 17grpid 14840 . . . 4  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (
(  .0.  ( +g  `  G )  .0.  )  =  .0.  <->  ( 0g `  G )  =  .0.  ) )
1912, 14, 18syl2anc 643 . . 3  |-  ( ph  ->  ( (  .0.  ( +g  `  G )  .0.  )  =  .0.  <->  ( 0g `  G )  =  .0.  ) )
2011, 19mpbid 202 . 2  |-  ( ph  ->  ( 0g `  G
)  =  .0.  )
2120eqcomd 2441 1  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685
This theorem is referenced by:  imasgrp2  14933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812
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