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Theorem grpinvval 14834
Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinvval.b  |-  B  =  ( Base `  G
)
grpinvval.p  |-  .+  =  ( +g  `  G )
grpinvval.o  |-  .0.  =  ( 0g `  G )
grpinvval.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvval  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
Distinct variable groups:    y, B    y, G    y, X
Allowed substitution hints:    .+ ( y)    N( y)    .0. ( y)

Proof of Theorem grpinvval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6081 . . . 4  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
21eqeq1d 2443 . . 3  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
32riotabidv 6543 . 2  |-  ( x  =  X  ->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  )  =  ( iota_ y  e.  B
( y  .+  X
)  =  .0.  )
)
4 grpinvval.b . . 3  |-  B  =  ( Base `  G
)
5 grpinvval.p . . 3  |-  .+  =  ( +g  `  G )
6 grpinvval.o . . 3  |-  .0.  =  ( 0g `  G )
7 grpinvval.n . . 3  |-  N  =  ( inv g `  G )
84, 5, 6, 7grpinvfval 14833 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
9 riotaex 6545 . 2  |-  ( iota_ y  e.  B ( y 
.+  X )  =  .0.  )  e.  _V
103, 8, 9fvmpt 5798 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13459   +g cplusg 13519   0gc0g 13713   inv gcminusg 14676
This theorem is referenced by:  grplinv  14841  isgrpinv  14845  xrsinvgval  24189  rnginvval  24218
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-riota 6541  df-minusg 14803
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