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Theorem grprinvlem 6287
 Description: Lemma for grprinvd 6288. (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 2791 . . . . 5
4 oveq2 6091 . . . . . . . 8
54eqeq1d 2446 . . . . . . 7
65rexbidv 2728 . . . . . 6
76cbvralv 2934 . . . . 5
83, 7sylib 190 . . . 4
9 oveq2 6091 . . . . . . 7
109eqeq1d 2446 . . . . . 6
1110rexbidv 2728 . . . . 5
1211rspccva 3053 . . . 4
138, 12sylan 459 . . 3
141, 13syldan 458 . 2
15 grprinvlem.e . . . . 5
1615oveq2d 6099 . . . 4
18 simprr 735 . . . . 5
1918oveq1d 6098 . . . 4
20 simpll 732 . . . . . 6
21 grprinvlem.a . . . . . . 7
2221caovassg 6247 . . . . . 6
2320, 22sylan 459 . . . . 5
24 simprl 734 . . . . 5
251adantr 453 . . . . 5
2623, 24, 25, 25caovassd 6248 . . . 4
27 grprinvlem.i . . . . . . . . 9
2827ralrimiva 2791 . . . . . . . 8
29 oveq2 6091 . . . . . . . . . 10
30 id 21 . . . . . . . . . 10
3129, 30eqeq12d 2452 . . . . . . . . 9
3231cbvralv 2934 . . . . . . . 8
3328, 32sylib 190 . . . . . . 7
3433adantr 453 . . . . . 6
35 oveq2 6091 . . . . . . . 8
36 id 21 . . . . . . . 8
3735, 36eqeq12d 2452 . . . . . . 7
3837rspcv 3050 . . . . . 6
391, 34, 38sylc 59 . . . . 5
4039adantr 453 . . . 4
4119, 26, 403eqtr3d 2478 . . 3
4217, 41, 183eqtr3d 2478 . 2
4314, 42rexlimddv 2836 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  wrex 2708  (class class class)co 6083 This theorem is referenced by:  grprinvd  6288 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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