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Theorem clmlmod 21751
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 2402 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2402 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
31, 2isclm 21748 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  (Scalar `  W )  =  (flds  (
Base `  (Scalar `  W
) ) )  /\  ( Base `  (Scalar `  W
) )  e.  (SubRing ` fld ) ) )
43simp1bi 1012 1  |-  ( W  e. CMod  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   ` cfv 5525  (class class class)co 6234   Basecbs 14733   ↾s cress 14734  Scalarcsca 14804  SubRingcsubrg 17637   LModclmod 17724  ℂfldccnfld 18632  CModcclm 21746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5489  df-fv 5533  df-ov 6237  df-clm 21747
This theorem is referenced by:  clmgrp  21752  clmabl  21753  clmring  21754  clmfgrp  21755  clmvsass  21771  clmvsdir  21772  clmvs1  21773  clm0vs  21774  clmvneg1  21775  clmvsneg  21776  clmsubdir  21778  zlmclm  21779  ttgbtwnid  24485  ttgcontlem1  24486
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