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Theorem grpridd 5725
 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 5547 . . . . . 6
32eqeq1d 2064 . . . . 5
43cbvrexv 2551 . . . 4
51, 4sylib 131 . . 3
6 grprinvlem.a . . . . . . . 8
76caovassg 5687 . . . . . . 7
87adantlr 454 . . . . . 6
9 simprl 491 . . . . . 6
10 simprrl 499 . . . . . 6
118, 9, 10, 9caovassd 5688 . . . . 5
12 grprinvlem.c . . . . . . 7
13 grprinvlem.o . . . . . . 7
14 grprinvlem.i . . . . . . 7
15 simprrr 500 . . . . . . 7
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 5724 . . . . . 6
1716oveq1d 5555 . . . . 5
1815oveq2d 5556 . . . . 5
1911, 17, 183eqtr3d 2096 . . . 4
2019anassrs 386 . . 3
215, 20rexlimddv 2454 . 2
2221, 14eqtr3d 2090 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   w3a 896   wceq 1259   wcel 1409  wrex 2324  (class class class)co 5540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543 This theorem is referenced by: (None)
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