Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  grprinvd Structured version   Unicode version

Theorem grprinvd 6286
 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 1154 . . . . 5
76caovclg 6239 . . . 4
9 grprinvd.x . . 3
10 grprinvd.n . . 3
118, 9, 10caovcld 6240 . 2
124caovassg 6245 . . . . 5
1312adantlr 696 . . . 4
1413, 9, 10, 11caovassd 6246 . . 3
15 grprinvd.e . . . . . 6
1615oveq1d 6096 . . . . 5
1713, 10, 9, 10caovassd 6246 . . . . 5
183ralrimiva 2789 . . . . . . . 8
19 oveq2 6089 . . . . . . . . . 10
20 id 20 . . . . . . . . . 10
2119, 20eqeq12d 2450 . . . . . . . . 9
2221cbvralv 2932 . . . . . . . 8
2318, 22sylib 189 . . . . . . 7
2423adantr 452 . . . . . 6
25 oveq2 6089 . . . . . . . 8
26 id 20 . . . . . . . 8
2725, 26eqeq12d 2450 . . . . . . 7
2827rspcv 3048 . . . . . 6
2910, 24, 28sylc 58 . . . . 5
3016, 17, 293eqtr3d 2476 . . . 4
3130oveq2d 6097 . . 3
3214, 31eqtrd 2468 . 2
331, 2, 3, 4, 5, 11, 32grprinvlem 6285 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706  (class class class)co 6081 This theorem is referenced by:  grpridd  6287  grprcan  14838  grprinv  14852 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
 Copyright terms: Public domain W3C validator