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Theorem grprinvlem 5695
 Description: Lemma for grprinvd 5696. (Contributed by NM, 9-Aug-2013.)
Hypotheses
Ref Expression
grprinvlem.c
grprinvlem.o
grprinvlem.i
grprinvlem.a
grprinvlem.n
grprinvlem.x
grprinvlem.e
Assertion
Ref Expression
grprinvlem
Distinct variable groups:   ,,,   ,,,   ,,,   , ,,   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem grprinvlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.x . . 3
2 grprinvlem.n . . . . . 6
32ralrimiva 2392 . . . . 5
4 oveq2 5520 . . . . . . . 8
54eqeq1d 2048 . . . . . . 7
65rexbidv 2327 . . . . . 6
76cbvralv 2533 . . . . 5
83, 7sylib 127 . . . 4
9 oveq2 5520 . . . . . . 7
109eqeq1d 2048 . . . . . 6
1110rexbidv 2327 . . . . 5
1211rspccva 2655 . . . 4
138, 12sylan 267 . . 3
141, 13syldan 266 . 2
15 grprinvlem.e . . . . 5
1615oveq2d 5528 . . . 4
1716adantr 261 . . 3
18 simprr 484 . . . . 5
1918oveq1d 5527 . . . 4
20 simpll 481 . . . . . 6
21 grprinvlem.a . . . . . . 7
2221caovassg 5659 . . . . . 6
2320, 22sylan 267 . . . . 5
24 simprl 483 . . . . 5
251adantr 261 . . . . 5
2623, 24, 25, 25caovassd 5660 . . . 4
27 grprinvlem.i . . . . . . . . 9
2827ralrimiva 2392 . . . . . . . 8
29 oveq2 5520 . . . . . . . . . 10
30 id 19 . . . . . . . . . 10
3129, 30eqeq12d 2054 . . . . . . . . 9
3231cbvralv 2533 . . . . . . . 8
3328, 32sylib 127 . . . . . . 7
3433adantr 261 . . . . . 6
35 oveq2 5520 . . . . . . . 8
36 id 19 . . . . . . . 8
3735, 36eqeq12d 2054 . . . . . . 7
3837rspcv 2652 . . . . . 6
391, 34, 38sylc 56 . . . . 5
4039adantr 261 . . . 4
4119, 26, 403eqtr3d 2080 . . 3
4217, 41, 183eqtr3d 2080 . 2
4314, 42rexlimddv 2437 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   w3a 885   wceq 1243   wcel 1393  wral 2306  wrex 2307  (class class class)co 5512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  grprinvd  5696
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