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Theorem lmhmclm 18600
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 15806 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
2 lmhmlmod2 15805 . . . 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 2296 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2296 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
64, 5lmhmsca 15803 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
76eqcomd 2301 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
87fveq2d 5545 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  T )
) )
98oveq2d 5890 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (flds  ( Base `  (Scalar `  S ) ) )  =  (flds  (
Base `  (Scalar `  T
) ) ) )
107, 9eqeq12d 2310 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( (Scalar `  S )  =  (flds  ( Base `  (Scalar `  S )
) )  <->  (Scalar `  T
)  =  (flds  ( Base `  (Scalar `  T ) ) ) ) )
118eleq1d 2362 . . 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 2296 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
144, 13isclm 18578 . 2  |-  ( S  e. CMod 
<->  ( S  e.  LMod  /\  (Scalar `  S )  =  (flds  (
Base `  (Scalar `  S
) ) )  /\  ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) ) )
15 eqid 2296 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
165, 15isclm 18578 . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165  Scalarcsca 13227  SubRingcsubrg 15557   LModclmod 15643   LMHom clmhm 15792  ℂfldccnfld 16393  CModcclm 18576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-lmhm 15795  df-clm 18577
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