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Theorem lmimgim 17489
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 17488 . . 3  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R LMHom  S ) )
2 lmghm 17455 . . 3  |-  ( F  e.  ( R LMHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
31, 2syl 16 . 2  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R  GrpHom  S ) )
4 eqid 2462 . . 3  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2462 . . 3  |-  ( Base `  S )  =  (
Base `  S )
64, 5lmimf1o 17487 . 2  |-  ( F  e.  ( R LMIso  S
)  ->  F :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
74, 5isgim 16100 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
83, 6, 7sylanbrc 664 1  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R GrpIso  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   Basecbs 14481    GrpHom cghm 16054   GrpIso cgim 16095   LMHom clmhm 17443   LMIso clmim 17444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-ghm 16055  df-gim 16097  df-lmhm 17446  df-lmim 17447
This theorem is referenced by: (None)
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