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Theorem islmim2 16169
Description: An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
islmim2  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  `' F  e.  ( S LMHom  R ) ) )

Proof of Theorem islmim2
StepHypRef Expression
1 eqid 2442 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2442 . . 3  |-  ( Base `  S )  =  (
Base `  S )
31, 2islmim 16165 . 2  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  F : (
Base `  R ) -1-1-onto-> ( Base `  S ) ) )
41, 2lmhmf1o 16153 . . 3  |-  ( F  e.  ( R LMHom  S
)  ->  ( F : ( Base `  R
)
-1-1-onto-> ( Base `  S )  <->  `' F  e.  ( S LMHom 
R ) ) )
54pm5.32i 620 . 2  |-  ( ( F  e.  ( R LMHom 
S )  /\  F : ( Base `  R
)
-1-1-onto-> ( Base `  S )
)  <->  ( F  e.  ( R LMHom  S )  /\  `' F  e.  ( S LMHom  R ) ) )
63, 5bitri 242 1  |-  ( F  e.  ( R LMIso  S
)  <->  ( F  e.  ( R LMHom  S )  /\  `' F  e.  ( S LMHom  R ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1727   `'ccnv 4906   -1-1-onto->wf1o 5482   ` cfv 5483  (class class class)co 6110   Basecbs 13500   LMHom clmhm 16126   LMIso clmim 16127
This theorem is referenced by:  lmimcnv  16170  lnmlmic  27201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-mnd 14721  df-grp 14843  df-ghm 15035  df-lmod 15983  df-lmhm 16129  df-lmim 16130
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