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

Theorem lmimcnv 16140
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 2437 . . . . . . . 8  |-  ( Base `  S )  =  (
Base `  S )
2 eqid 2437 . . . . . . . 8  |-  ( Base `  T )  =  (
Base `  T )
31, 2lmhmf 16111 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
4 frel 5595 . . . . . . 7  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Rel  F )
53, 4syl 16 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  Rel  F )
6 dfrel2 5322 . . . . . 6  |-  ( Rel 
F  <->  `' `' F  =  F
)
75, 6sylib 190 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  `' `' F  =  F )
8 id 21 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S LMHom  T ) )
97, 8eqeltrd 2511 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  `' `' F  e.  ( S LMHom  T ) )
109anim2i 554 . . 3  |-  ( ( `' F  e.  ( T LMHom  S )  /\  F  e.  ( S LMHom  T ) )  ->  ( `' F  e.  ( T LMHom  S )  /\  `' `' F  e.  ( S LMHom  T ) ) )
1110ancoms 441 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  `' F  e.  ( T LMHom  S ) )  ->  ( `' F  e.  ( T LMHom  S )  /\  `' `' F  e.  ( S LMHom  T ) ) )
12 islmim2 16139 . 2  |-  ( F  e.  ( S LMIso  T
)  <->  ( F  e.  ( S LMHom  T )  /\  `' F  e.  ( T LMHom  S ) ) )
13 islmim2 16139 . 2  |-  ( `' F  e.  ( T LMIso 
S )  <->  ( `' F  e.  ( T LMHom  S )  /\  `' `' F  e.  ( S LMHom  T ) ) )
1411, 12, 133imtr4i 259 1  |-  ( F  e.  ( S LMIso  T
)  ->  `' F  e.  ( T LMIso  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   `'ccnv 4878   Rel wrel 4884   -->wf 5451   ` cfv 5455  (class class class)co 6082   Basecbs 13470   LMHom clmhm 16096   LMIso clmim 16097
This theorem is referenced by:  lmicsym  16145  lbslcic  27289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-mnd 14691  df-grp 14813  df-ghm 15005  df-lmod 15953  df-lmhm 16099  df-lmim 16100
  Copyright terms: Public domain W3C validator