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Theorem grpridd 6510
 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 6302 . . . . . 6
32eqeq1d 2469 . . . . 5
43cbvrexv 3094 . . . 4
51, 4sylib 196 . . 3
6 grprinvlem.a . . . . . . . 8
76caovassg 6468 . . . . . . 7
87adantlr 714 . . . . . 6
9 simprl 755 . . . . . 6
10 simprrl 763 . . . . . 6
118, 9, 10, 9caovassd 6469 . . . . 5
12 grprinvlem.c . . . . . . 7
13 grprinvlem.o . . . . . . 7
14 grprinvlem.i . . . . . . 7
15 simprrr 764 . . . . . . 7
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 6509 . . . . . 6
1716oveq1d 6310 . . . . 5
1815oveq2d 6311 . . . . 5
1911, 17, 183eqtr3d 2516 . . . 4
2019anassrs 648 . . 3
215, 20rexlimddv 2963 . 2
2221, 14eqtr3d 2510 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  wrex 2818  (class class class)co 6295 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298 This theorem is referenced by:  isgrpde  15946
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