MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmimgim Structured version   Unicode version

Theorem lmimgim 16129
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 16128 . . 3  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R LMHom  S ) )
2 lmghm 16099 . . 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 2435 . . 3  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2435 . . 3  |-  ( Base `  S )  =  (
Base `  S )
64, 5lmimf1o 16127 . 2  |-  ( F  e.  ( R LMIso  S
)  ->  F :
( Base `  R ) -1-1-onto-> ( Base `  S ) )
74, 5isgim 15041 . 2  |-  ( F  e.  ( R GrpIso  S
)  <->  ( F  e.  ( R  GrpHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
83, 6, 7sylanbrc 646 1  |-  ( F  e.  ( R LMIso  S
)  ->  F  e.  ( R GrpIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Basecbs 13461    GrpHom cghm 14995   GrpIso cgim 15036   LMHom clmhm 16087   LMIso clmim 16088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ghm 14996  df-gim 15038  df-lmhm 16090  df-lmim 16091
  Copyright terms: Public domain W3C validator