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Theorem grpoinvval 20908
 Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1
grpinvfval.2 GId
grpinvfval.3
Assertion
Ref Expression
grpoinvval
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem grpoinvval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.1 . . . 4
2 grpinvfval.2 . . . 4 GId
3 grpinvfval.3 . . . 4
41, 2, 3grpoinvfval 20907 . . 3
54fveq1d 5543 . 2
6 oveq2 5882 . . . . 5
76eqeq1d 2304 . . . 4
87riotabidv 6322 . . 3
9 eqid 2296 . . 3
10 riotaex 6324 . . 3
118, 9, 10fvmpt 5618 . 2
125, 11sylan9eq 2348 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696   cmpt 4093   crn 4706  cfv 5271  (class class class)co 5874  crio 6313  cgr 20869  GIdcgi 20870  cgn 20871 This theorem is referenced by:  grpoinvcl  20909  grpoinv  20910  addinv  21035 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-ginv 20876
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