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Theorem clmlmod 21302
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 2467 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2467 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
31, 2isclm 21299 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  (Scalar `  W )  =  (flds  (
Base `  (Scalar `  W
) ) )  /\  ( Base `  (Scalar `  W
) )  e.  (SubRing ` fld ) ) )
43simp1bi 1011 1  |-  ( W  e. CMod  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   ↾s cress 14487  Scalarcsca 14554  SubRingcsubrg 17208   LModclmod 17295  ℂfldccnfld 18191  CModcclm 21297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-clm 21298
This theorem is referenced by:  clmgrp  21303  clmabl  21304  clmrng  21305  clmfgrp  21306  clmvsass  21322  clmvsdir  21323  clmvs1  21324  clm0vs  21325  clmvneg1  21326  clmvsneg  21327  clmsubdir  21329  zlmclm  21330  ttgbtwnid  23863  ttgcontlem1  23864
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