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Theorem lmhmclm 18584
Description: The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
Assertion
Ref Expression
lmhmclm  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e. CMod  <-> 
T  e. CMod ) )

Proof of Theorem lmhmclm
StepHypRef Expression
1 lmhmlmod1 15790 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
2 lmhmlmod2 15789 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
31, 22thd 231 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e.  LMod  <->  T  e.  LMod ) )
4 eqid 2283 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2283 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
64, 5lmhmsca 15787 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
76eqcomd 2288 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
87fveq2d 5529 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  T )
) )
98oveq2d 5874 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (flds  ( Base `  (Scalar `  S ) ) )  =  (flds  (
Base `  (Scalar `  T
) ) ) )
107, 9eqeq12d 2297 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( (Scalar `  S )  =  (flds  ( Base `  (Scalar `  S )
) )  <->  (Scalar `  T
)  =  (flds  ( Base `  (Scalar `  T ) ) ) ) )
118eleq1d 2349 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) 
<->  ( Base `  (Scalar `  T ) )  e.  (SubRing ` fld ) ) )
123, 10, 113anbi123d 1252 . 2  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( S  e.  LMod  /\  (Scalar `  S )  =  (flds  ( Base `  (Scalar `  S )
) )  /\  ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) )  <->  ( T  e.  LMod  /\  (Scalar `  T
)  =  (flds  ( Base `  (Scalar `  T ) ) )  /\  ( Base `  (Scalar `  T ) )  e.  (SubRing ` fld ) ) ) )
13 eqid 2283 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
144, 13isclm 18562 . 2  |-  ( S  e. CMod 
<->  ( S  e.  LMod  /\  (Scalar `  S )  =  (flds  (
Base `  (Scalar `  S
) ) )  /\  ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) ) )
15 eqid 2283 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
165, 15isclm 18562 . 2  |-  ( T  e. CMod 
<->  ( T  e.  LMod  /\  (Scalar `  T )  =  (flds  (
Base `  (Scalar `  T
) ) )  /\  ( Base `  (Scalar `  T
) )  e.  (SubRing ` fld ) ) )
1712, 14, 163bitr4g 279 1  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e. CMod  <-> 
T  e. CMod ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211  SubRingcsubrg 15541   LModclmod 15627   LMHom clmhm 15776  ℂfldccnfld 16377  CModcclm 18560
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
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-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-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-lmhm 15779  df-clm 18561
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