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Theorem grprinvd 6513
 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 1206 . . . . 5
76caovclg 6466 . . . 4
87adantlr 719 . . 3
9 grprinvd.x . . 3
10 grprinvd.n . . 3
118, 9, 10caovcld 6467 . 2
124caovassg 6472 . . . . 5
1312adantlr 719 . . . 4
1413, 9, 10, 11caovassd 6473 . . 3
15 grprinvd.e . . . . . 6
1615oveq1d 6311 . . . . 5
1713, 10, 9, 10caovassd 6473 . . . . 5
183ralrimiva 2837 . . . . . . . 8
19 oveq2 6304 . . . . . . . . . 10
20 id 23 . . . . . . . . . 10
2119, 20eqeq12d 2442 . . . . . . . . 9
2221cbvralv 3053 . . . . . . . 8
2318, 22sylib 199 . . . . . . 7
2423adantr 466 . . . . . 6
25 oveq2 6304 . . . . . . . 8
26 id 23 . . . . . . . 8
2725, 26eqeq12d 2442 . . . . . . 7
2827rspcv 3175 . . . . . 6
2910, 24, 28sylc 62 . . . . 5
3016, 17, 293eqtr3d 2469 . . . 4
3130oveq2d 6312 . . 3
3214, 31eqtrd 2461 . 2
331, 2, 3, 4, 5, 11, 32grprinvlem 6512 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1867  wral 2773  wrex 2774  (class class class)co 6296 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299 This theorem is referenced by:  grpridd  6514  grprcan  16643  grprinv  16657
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