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Theorem grprinvlem 6512
 Description: Lemma for grprinvd 6513. (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 2804 . . . . 5
4 oveq2 6303 . . . . . . . 8
54eqeq1d 2455 . . . . . . 7
65rexbidv 2903 . . . . . 6
76cbvralv 3021 . . . . 5
83, 7sylib 200 . . . 4
9 oveq2 6303 . . . . . . 7
109eqeq1d 2455 . . . . . 6
1110rexbidv 2903 . . . . 5
1211rspccva 3151 . . . 4
138, 12sylan 474 . . 3
141, 13syldan 473 . 2
15 grprinvlem.e . . . . 5
1615oveq2d 6311 . . . 4
18 simprr 767 . . . . 5
1918oveq1d 6310 . . . 4
20 simpll 761 . . . . . 6
21 grprinvlem.a . . . . . . 7
2221caovassg 6472 . . . . . 6
2320, 22sylan 474 . . . . 5
24 simprl 765 . . . . 5
251adantr 467 . . . . 5
2623, 24, 25, 25caovassd 6473 . . . 4
27 grprinvlem.i . . . . . . . . 9
2827ralrimiva 2804 . . . . . . . 8
29 oveq2 6303 . . . . . . . . . 10
30 id 22 . . . . . . . . . 10
3129, 30eqeq12d 2468 . . . . . . . . 9
3231cbvralv 3021 . . . . . . . 8
3328, 32sylib 200 . . . . . . 7
3433adantr 467 . . . . . 6
35 oveq2 6303 . . . . . . . 8
36 id 22 . . . . . . . 8
3735, 36eqeq12d 2468 . . . . . . 7
3837rspcv 3148 . . . . . 6
391, 34, 38sylc 62 . . . . 5
4039adantr 467 . . . 4
4119, 26, 403eqtr3d 2495 . . 3
4217, 41, 183eqtr3d 2495 . 2
4314, 42rexlimddv 2885 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   w3a 986   wceq 1446   wcel 1889  wral 2739  wrex 2740  (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 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-iota 5549  df-fv 5593  df-ov 6298 This theorem is referenced by:  grprinvd  6513
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