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Theorem clmlmod 22861
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 2621 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2621 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 22858 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1075 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  cfv 5886  (class class class)co 6647  Basecbs 15851  s cress 15852  Scalarcsca 15938  SubRingcsubrg 18770  LModclmod 18857  fldccnfld 19740  ℂModcclm 22856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-clm 22857
This theorem is referenced by:  clmgrp  22862  clmabl  22863  clmring  22864  clmfgrp  22865  clmvscl  22882  clmvsass  22883  clmvsdir  22885  clmvsdi  22886  clmvs1  22887  clmvs2  22888  clm0vs  22889  clmopfne  22890  clmvneg1  22893  clmvsneg  22894  clmsubdir  22896  clmvsubval  22903  zlmclm  22906  cmodscmulexp  22916  iscvs  22921  cvsi  22924  isncvsngp  22943  ttgbtwnid  25758  ttgcontlem1  25759
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