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Theorem clmlmod 20644
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 2443 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2443 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
31, 2isclm 20641 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  (Scalar `  W )  =  (flds  (
Base `  (Scalar `  W
) ) )  /\  ( Base `  (Scalar `  W
) )  e.  (SubRing ` fld ) ) )
43simp1bi 1003 1  |-  ( W  e. CMod  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   Basecbs 14179   ↾s cress 14180  Scalarcsca 14246  SubRingcsubrg 16866   LModclmod 16953  ℂfldccnfld 17823  CModcclm 20639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4426
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099  df-clm 20640
This theorem is referenced by:  clmgrp  20645  clmabl  20646  clmrng  20647  clmfgrp  20648  clmvsass  20664  clmvsdir  20665  clmvs1  20666  clm0vs  20667  clmvneg1  20668  clmvsneg  20669  clmsubdir  20671  zlmclm  20672  ttgbtwnid  23135  ttgcontlem1  23136
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