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Theorem grpoid 21803
Description: Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinveu.1  |-  X  =  ran  G
grpinveu.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoid  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )

Proof of Theorem grpoid
StepHypRef Expression
1 grpinveu.1 . . . . . 6  |-  X  =  ran  G
2 grpinveu.2 . . . . . 6  |-  U  =  (GId `  G )
31, 2grpoidcl 21797 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  X )
41grporcan 21801 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  U  e.  X  /\  A  e.  X )
)  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U
) )
543exp2 1171 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( U  e.  X  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) ) ) )
63, 5mpid 39 . . . 4  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( A  e.  X  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) ) ) )
76pm2.43d 46 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) )
87imp 419 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) )
91, 2grpolid 21799 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
109eqeq2d 2446 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  ( A G A )  =  A ) )
118, 10bitr3d 247 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   GrpOpcgr 21766  GIdcgi 21767
This theorem is referenced by:  subgoid  21887  ghomid  21945  hhssnv  22756  ghomgrpilem2  25089
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-grpo 21771  df-gid 21772
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