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Theorem lmimcnv 15836
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 2296 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2296 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 15807 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5408 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
53, 4syl 15 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  Rel  F )
6 dfrel2 5140 . . . . . 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 2370 . . . 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 15835 . 2  |-  ( F  e.  ( S LMIso  T
)  <->  ( F  e.  ( S LMHom  T )  /\  `' F  e.  ( T LMHom  S ) ) )
13 islmim2 15835 . 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 1632    e. wcel 1696   `'ccnv 4704   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   LMHom clmhm 15792   LMIso clmim 15793
This theorem is referenced by:  lmicsym  15841  lbslcic  27414
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-mnd 14383  df-grp 14505  df-ghm 14697  df-lmod 15645  df-lmhm 15795  df-lmim 15796
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