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Theorem grpinvval 15890
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  =  ( invg `  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 6290 . . . 4  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
21eqeq1d 2469 . . 3  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
32riotabidv 6245 . 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  =  ( invg `  G )
84, 5, 6, 7grpinvfval 15889 . 2  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B  ( y  .+  x
)  =  .0.  )
)
9 riotaex 6247 . 2  |-  ( iota_ y  e.  B  ( y 
.+  X )  =  .0.  )  e.  _V
103, 8, 9fvmpt 5948 1  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B  ( y  .+  X )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   invgcminusg 15724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-minusg 15859
This theorem is referenced by:  grplinv  15897  isgrpinv  15901  xrsinvgval  27327  rnginvval  27445
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