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Theorem grprinvd 5696
 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 1105 . . . . 5
76caovclg 5653 . . . 4
9 grprinvd.x . . 3
10 grprinvd.n . . 3
118, 9, 10caovcld 5654 . 2
124caovassg 5659 . . . . 5
1312adantlr 446 . . . 4
1413, 9, 10, 11caovassd 5660 . . 3
15 grprinvd.e . . . . . 6
1615oveq1d 5527 . . . . 5
1713, 10, 9, 10caovassd 5660 . . . . 5
183ralrimiva 2392 . . . . . . . 8
19 oveq2 5520 . . . . . . . . . 10
20 id 19 . . . . . . . . . 10
2119, 20eqeq12d 2054 . . . . . . . . 9
2221cbvralv 2533 . . . . . . . 8
2318, 22sylib 127 . . . . . . 7
2423adantr 261 . . . . . 6
25 oveq2 5520 . . . . . . . 8
26 id 19 . . . . . . . 8
2725, 26eqeq12d 2054 . . . . . . 7
2827rspcv 2652 . . . . . 6
2910, 24, 28sylc 56 . . . . 5
3016, 17, 293eqtr3d 2080 . . . 4
3130oveq2d 5528 . . 3
3214, 31eqtrd 2072 . 2
331, 2, 3, 4, 5, 11, 32grprinvlem 5695 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   w3a 885   wceq 1243   wcel 1393  wral 2306  wrex 2307  (class class class)co 5512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  grpridd  5697
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