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Theorem lmhmclm 19104
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 16102 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
2 lmhmlmod2 16101 . . . 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 2436 . . . . . 6  |-  (Scalar `  S )  =  (Scalar `  S )
5 eqid 2436 . . . . . 6  |-  (Scalar `  T )  =  (Scalar `  T )
64, 5lmhmsca 16099 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
76eqcomd 2441 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  S
)  =  (Scalar `  T ) )
87fveq2d 5725 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  T )
) )
98oveq2d 6090 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (flds  ( Base `  (Scalar `  S ) ) )  =  (flds  (
Base `  (Scalar `  T
) ) ) )
107, 9eqeq12d 2450 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  ( (Scalar `  S )  =  (flds  ( Base `  (Scalar `  S )
) )  <->  (Scalar `  T
)  =  (flds  ( Base `  (Scalar `  T ) ) ) ) )
118eleq1d 2502 . . 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 2436 . . 3  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
144, 13isclm 19082 . 2  |-  ( S  e. CMod 
<->  ( S  e.  LMod  /\  (Scalar `  S )  =  (flds  (
Base `  (Scalar `  S
) ) )  /\  ( Base `  (Scalar `  S
) )  e.  (SubRing ` fld ) ) )
15 eqid 2436 . . 3  |-  ( Base `  (Scalar `  T )
)  =  ( Base `  (Scalar `  T )
)
165, 15isclm 19082 . 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 1652    e. wcel 1725   ` cfv 5447  (class class class)co 6074   Basecbs 13462   ↾s cress 13463  Scalarcsca 13525  SubRingcsubrg 15857   LModclmod 15943   LMHom clmhm 16088  ℂfldccnfld 16696  CModcclm 19080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-lmhm 16091  df-clm 19081
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