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Theorem grpridd 6289
 Description: Deduce right identity 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
Assertion
Ref Expression
grpridd
Distinct variable groups:   ,,,   ,,,   ,,,   , ,,

Proof of Theorem grpridd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.n . . . 4
2 oveq1 6090 . . . . . 6
32eqeq1d 2446 . . . . 5
43cbvrexv 2935 . . . 4
51, 4sylib 190 . . 3
6 grprinvlem.a . . . . . . . 8
76caovassg 6247 . . . . . . 7
87adantlr 697 . . . . . 6
9 simprl 734 . . . . . 6
10 simprrl 742 . . . . . 6
118, 9, 10, 9caovassd 6248 . . . . 5
12 grprinvlem.c . . . . . . 7
13 grprinvlem.o . . . . . . 7
14 grprinvlem.i . . . . . . 7
15 simprrr 743 . . . . . . 7
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 6288 . . . . . 6
1716oveq1d 6098 . . . . 5
1815oveq2d 6099 . . . . 5
1911, 17, 183eqtr3d 2478 . . . 4
2019anassrs 631 . . 3
215, 20rexlimddv 2836 . 2
2221, 14eqtr3d 2472 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wrex 2708  (class class class)co 6083 This theorem is referenced by:  isgrpde  14831 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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