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Theorem brlmic 15821
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 15781 . 2  |-  ~=ph𝑚  =  ( `' LMIso  " ( _V  \  1o ) )
2 lmimfn 15783 . 2  |- LMIso  Fn  ( LMod  X.  LMod )
31, 2brwitnlem 6506 1  |-  ( R 
~=ph𝑚  S 
<->  ( R LMIso  S )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    =/= wne 2446   (/)c0 3455   class class class wbr 4023    X. cxp 4687  (class class class)co 5858   LModclmod 15627   LMIso clmim 15777    ~=ph𝑚 clmic 15778
This theorem is referenced by:  brlmici  15822  lmiclcl  15823  lmicrcl  15824  lmicsym  15825  lnmlmic  27186  lmiclbs  27307
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-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-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-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-suc 4398  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-lmim 15780  df-lmic 15781
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