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Theorem lmimgim 15834
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 15833 . . 3  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R LMHom  S ) )
2 lmghm 15804 . . 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 2296 . . 3  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2296 . . 3  |-  ( Base `  S )  =  (
Base `  S )
64, 5lmimf1o 15832 . 2  |-  ( F  e.  ( R LMIso  S
)  ->  F :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
74, 5isgim 14742 . 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 1696   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164    GrpHom cghm 14696   GrpIso cgim 14737   LMHom clmhm 15792   LMIso clmim 15793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ghm 14697  df-gim 14739  df-lmhm 15795  df-lmim 15796
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