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Theorem lmhmclm 18975
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 16029 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
2 lmhmlmod2 16028 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
31, 22thd 232 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e.  LMod  <->  T  e.  LMod ) )
4 eqid 2380 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2380 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
64, 5lmhmsca 16026 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
76eqcomd 2385 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
87fveq2d 5665 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  T )
) )
98oveq2d 6029 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (flds  ( Base `  (Scalar `  S ) ) )  =  (flds  (
Base `  (Scalar `  T
) ) ) )
107, 9eqeq12d 2394 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( (Scalar `  S )  =  (flds  ( Base `  (Scalar `  S )
) )  <->  (Scalar `  T
)  =  (flds  ( Base `  (Scalar `  T ) ) ) ) )
118eleq1d 2446 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) 
<->  ( Base `  (Scalar `  T ) )  e.  (SubRing ` fld ) ) )
123, 10, 113anbi123d 1254 . 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 2380 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
144, 13isclm 18953 . 2  |-  ( S  e. CMod 
<->  ( S  e.  LMod  /\  (Scalar `  S )  =  (flds  (
Base `  (Scalar `  S
) ) )  /\  ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) ) )
15 eqid 2380 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
165, 15isclm 18953 . 2  |-  ( T  e. CMod 
<->  ( T  e.  LMod  /\  (Scalar `  T )  =  (flds  (
Base `  (Scalar `  T
) ) )  /\  ( Base `  (Scalar `  T
) )  e.  (SubRing ` fld ) ) )
1712, 14, 163bitr4g 280 1  |-  ( F  e.  ( S LMHom  T
)  ->  ( S  e. CMod  <-> 
T  e. CMod ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   ↾s cress 13390  Scalarcsca 13452  SubRingcsubrg 15784   LModclmod 15870   LMHom clmhm 16015  ℂfldccnfld 16619  CModcclm 18951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-lmhm 16018  df-clm 18952
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