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Theorem grprinvd 5959
 Description: Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grprinvlem.c
grprinvlem.o
grprinvlem.i
grprinvlem.a
grprinvlem.n
grprinvd.x
grprinvd.n
grprinvd.e
Assertion
Ref Expression
grprinvd
Distinct variable groups:   ,,,   ,,,   ,,,   ,,   , ,,   ,,   ,
Allowed substitution hints:   (,)   ()   ()

Proof of Theorem grprinvd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.c . 2
2 grprinvlem.o . 2
3 grprinvlem.i . 2
4 grprinvlem.a . 2
5 grprinvlem.n . 2
613expb 1182 . . . . 5
76caovclg 5916 . . . 4
9 grprinvd.x . . 3
10 grprinvd.n . . 3
118, 9, 10caovcld 5917 . 2
124caovassg 5922 . . . . 5
1312adantlr 468 . . . 4
1413, 9, 10, 11caovassd 5923 . . 3
15 grprinvd.e . . . . . 6
1615oveq1d 5782 . . . . 5
1713, 10, 9, 10caovassd 5923 . . . . 5
18 oveq2 5775 . . . . . . 7
19 id 19 . . . . . . 7
2018, 19eqeq12d 2152 . . . . . 6
213ralrimiva 2503 . . . . . . . 8
22 oveq2 5775 . . . . . . . . . 10
23 id 19 . . . . . . . . . 10
2422, 23eqeq12d 2152 . . . . . . . . 9
2524cbvralv 2652 . . . . . . . 8
2621, 25sylib 121 . . . . . . 7
2726adantr 274 . . . . . 6
2820, 27, 10rspcdva 2789 . . . . 5
2916, 17, 283eqtr3d 2178 . . . 4
3029oveq2d 5783 . . 3
3114, 30eqtrd 2170 . 2
321, 2, 3, 4, 5, 11, 31grprinvlem 5958 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   w3a 962   wceq 1331   wcel 1480  wral 2414  wrex 2415  (class class class)co 5767 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770 This theorem is referenced by:  grpridd  5960
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