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Theorem clmlmod 25114
Description: A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Assertion
Ref Expression
clmlmod (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)

Proof of Theorem clmlmod
StepHypRef Expression
1 eqid 2735 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2735 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 25111 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1144 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Scalarcsca 17301  SubRingcsubrg 20586  LModclmod 20875  fldccnfld 21382  ℂModcclm 25109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-clm 25110
This theorem is referenced by:  clmgrp  25115  clmabl  25116  clmring  25117  clmfgrp  25118  clmvscl  25135  clmvsass  25136  clmvsdir  25138  clmvsdi  25139  clmvs1  25140  clmvs2  25141  clm0vs  25142  clmopfne  25143  clmvneg1  25146  clmvsneg  25147  clmsubdir  25149  clmvsubval  25156  zlmclm  25159  cmodscmulexp  25169  iscvs  25174  cvsi  25177  isncvsngp  25197  ttgbtwnid  28913  ttgcontlem1  28914
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