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Theorem lmiclcl 15916
Description: Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
lmiclcl  |-  ( R 
~=ph𝑚  S  ->  R  e.  LMod )

Proof of Theorem lmiclcl
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brlmic 15914 . . 3  |-  ( R 
~=ph𝑚  S 
<->  ( R LMIso  S )  =/=  (/) )
2 n0 3540 . . 3  |-  ( ( R LMIso  S )  =/=  (/) 
<->  E. f  f  e.  ( R LMIso  S ) )
31, 2bitri 240 . 2  |-  ( R 
~=ph𝑚  S 
<->  E. f  f  e.  ( R LMIso  S ) )
4 lmimlmhm 15910 . . . 4  |-  ( f  e.  ( R LMIso  S
)  ->  f  e.  ( R LMHom  S ) )
5 lmhmlmod1 15883 . . . 4  |-  ( f  e.  ( R LMHom  S
)  ->  R  e.  LMod )
64, 5syl 15 . . 3  |-  ( f  e.  ( R LMIso  S
)  ->  R  e.  LMod )
76exlimiv 1634 . 2  |-  ( E. f  f  e.  ( R LMIso  S )  ->  R  e.  LMod )
83, 7sylbi 187 1  |-  ( R 
~=ph𝑚  S  ->  R  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1541    e. wcel 1710    =/= wne 2521   (/)c0 3531   class class class wbr 4102  (class class class)co 5942   LModclmod 15720   LMHom clmhm 15869   LMIso clmim 15870    ~=ph𝑚 clmic 15871
This theorem is referenced by:  lmisfree  26635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-suc 4477  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-1o 6563  df-lmhm 15872  df-lmim 15873  df-lmic 15874
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