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Theorem clmlmod 19045
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 2404 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2404 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
31, 2isclm 19042 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  (Scalar `  W )  =  (flds  (
Base `  (Scalar `  W
) ) )  /\  ( Base `  (Scalar `  W
) )  e.  (SubRing ` fld ) ) )
43simp1bi 972 1  |-  ( W  e. CMod  ->  W  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487  SubRingcsubrg 15819   LModclmod 15905  ℂfldccnfld 16658  CModcclm 19040
This theorem is referenced by:  clmgrp  19046  clmabl  19047  clmrng  19048  clmfgrp  19049  clmvsass  19065  clmvsdir  19066  clmvs1  19067  clm0vs  19068  clmvneg1  19069  clmvsneg  19070  clmsubdir  19072  zlmclm  19073
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
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 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-clm 19041
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