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Theorem brlmic 16140
Description: The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
brlmic  |-  ( R 
~=ph𝑚  S 
<->  ( R LMIso  S )  =/=  (/) )

Proof of Theorem brlmic
StepHypRef Expression
1 df-lmic 16100 . 2  |-  ~=ph𝑚  =  ( `' LMIso  " ( _V  \  1o ) )
2 lmimfn 16102 . 2  |- LMIso  Fn  ( LMod  X.  LMod )
31, 2brwitnlem 6751 1  |-  ( R 
~=ph𝑚  S 
<->  ( R LMIso  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    =/= wne 2599   (/)c0 3628   class class class wbr 4212    X. cxp 4876  (class class class)co 6081   LModclmod 15950   LMIso clmim 16096    ~=ph𝑚 clmic 16097
This theorem is referenced by:  brlmici  16141  lmiclcl  16142  lmicrcl  16143  lmicsym  16144  lnmlmic  27163  lmiclbs  27284
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-1o 6724  df-lmim 16099  df-lmic 16100
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