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Theorem grpridd 6102
 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 5907 . . . . . 6
32eqeq1d 2324 . . . . 5
43cbvrexv 2799 . . . 4
51, 4sylib 188 . . 3
6 grprinvlem.a . . . . . . . . . 10
76caovassg 6060 . . . . . . . . 9
87adantlr 695 . . . . . . . 8
9 simprl 732 . . . . . . . 8
10 simprrl 740 . . . . . . . 8
118, 9, 10, 9caovassd 6061 . . . . . . 7
12 grprinvlem.c . . . . . . . . 9
13 grprinvlem.o . . . . . . . . 9
14 grprinvlem.i . . . . . . . . 9
15 simprrr 741 . . . . . . . . 9
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 6101 . . . . . . . 8
1716oveq1d 5915 . . . . . . 7
1815oveq2d 5916 . . . . . . 7
1911, 17, 183eqtr3d 2356 . . . . . 6
2019anassrs 629 . . . . 5
2120expr 598 . . . 4
2221rexlimdva 2701 . . 3
235, 22mpd 14 . 2
2423, 14eqtr3d 2350 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1633   wcel 1701  wrex 2578  (class class class)co 5900 This theorem is referenced by:  isgrpde  14555 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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