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Theorem clmlmod 22110
Description: A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Assertion
Ref Expression
clmlmod  |-  ( W  e. CMod  ->  W  e.  LMod )

Proof of Theorem clmlmod
StepHypRef Expression
1 eqid 2453 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2453 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
31, 2isclm 22107 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  (Scalar `  W )  =  (flds  (
Base `  (Scalar `  W
) ) )  /\  ( Base `  (Scalar `  W
) )  e.  (SubRing ` fld ) ) )
43simp1bi 1024 1  |-  ( W  e. CMod  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446    e. wcel 1889   ` cfv 5585  (class class class)co 6295   Basecbs 15133   ↾s cress 15134  Scalarcsca 15205  SubRingcsubrg 18016   LModclmod 18103  ℂfldccnfld 18982  CModcclm 22105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-nul 4537
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-iota 5549  df-fv 5593  df-ov 6298  df-clm 22106
This theorem is referenced by:  clmgrp  22111  clmabl  22112  clmring  22113  clmfgrp  22114  clmvsass  22130  clmvsdir  22131  clmvs1  22132  clm0vs  22133  clmvneg1  22134  clmvsneg  22135  clmsubdir  22137  zlmclm  22138  ttgbtwnid  24926  ttgcontlem1  24927
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