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Theorem grpoid 23722
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 23716 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  X )
41grporcan 23720 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  U  e.  X  /\  A  e.  X )
)  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U
) )
543exp2 1205 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( U  e.  X  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) ) ) )
63, 5mpid 41 . . . 4  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( A  e.  X  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) ) ) )
76pm2.43d 48 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) )
87imp 429 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) )
91, 2grpolid 23718 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
109eqeq2d 2454 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  ( A G A )  =  A ) )
118, 10bitr3d 255 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ran crn 4853   ` cfv 5430  (class class class)co 6103   GrpOpcgr 23685  GIdcgi 23686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438  df-riota 6064  df-ov 6106  df-grpo 23690  df-gid 23691
This theorem is referenced by:  subgoid  23806  ghomid  23864  hhssnv  24677  ghomgrpilem2  27317
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