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Theorem lmimgim 15818
Description: An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
lmimgim  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R GrpIso  S ) )

Proof of Theorem lmimgim
StepHypRef Expression
1 lmimlmhm 15817 . . 3  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R LMHom  S ) )
2 lmghm 15788 . . 3  |-  ( F  e.  ( R LMHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
31, 2syl 15 . 2  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R  GrpHom  S ) )
4 eqid 2283 . . 3  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2283 . . 3  |-  ( Base `  S )  =  (
Base `  S )
64, 5lmimf1o 15816 . 2  |-  ( F  e.  ( R LMIso  S
)  ->  F :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
74, 5isgim 14726 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
83, 6, 7sylanbrc 645 1  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R GrpIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148    GrpHom cghm 14680   GrpIso cgim 14721   LMHom clmhm 15776   LMIso clmim 15777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ghm 14681  df-gim 14723  df-lmhm 15779  df-lmim 15780
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