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Theorem lmimcnv 15820
Description: The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
lmimcnv  |-  ( F  e.  ( S LMIso  T
)  ->  `' F  e.  ( T LMIso  S ) )

Proof of Theorem lmimcnv
StepHypRef Expression
1 eqid 2283 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2283 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 15791 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5392 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
53, 4syl 15 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  Rel  F )
6 dfrel2 5124 . . . . . 6  |-  ( Rel 
F  <->  `' `' F  =  F
)
75, 6sylib 188 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  `' `' F  =  F )
8 id 19 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S LMHom  T ) )
97, 8eqeltrd 2357 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  `' `' F  e.  ( S LMHom  T ) )
109anim2i 552 . . 3  |-  ( ( `' F  e.  ( T LMHom  S )  /\  F  e.  ( S LMHom  T ) )  ->  ( `' F  e.  ( T LMHom  S )  /\  `' `' F  e.  ( S LMHom  T ) ) )
1110ancoms 439 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  ( `' F  e.  ( T LMHom  S )  /\  `' `' F  e.  ( S LMHom  T ) ) )
12 islmim2 15819 . 2  |-  ( F  e.  ( S LMIso  T
)  <->  ( F  e.  ( S LMHom  T )  /\  `' F  e.  ( T LMHom  S ) ) )
13 islmim2 15819 . 2  |-  ( `' F  e.  ( T LMIso 
S )  <->  ( `' F  e.  ( T LMHom  S )  /\  `' `' F  e.  ( S LMHom  T ) ) )
1411, 12, 133imtr4i 257 1  |-  ( F  e.  ( S LMIso  T
)  ->  `' F  e.  ( T LMIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   `'ccnv 4688   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   LMHom clmhm 15776   LMIso clmim 15777
This theorem is referenced by:  lmicsym  15825  lbslcic  27311
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-mnd 14367  df-grp 14489  df-ghm 14681  df-lmod 15629  df-lmhm 15779  df-lmim 15780
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