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Theorem clmlmod 24583
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 2733 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2733 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 24580 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1146 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6544  (class class class)co 7409  Basecbs 17144  s cress 17173  Scalarcsca 17200  SubRingcsubrg 20315  LModclmod 20471  fldccnfld 20944  ℂModcclm 24578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-clm 24579
This theorem is referenced by:  clmgrp  24584  clmabl  24585  clmring  24586  clmfgrp  24587  clmvscl  24604  clmvsass  24605  clmvsdir  24607  clmvsdi  24608  clmvs1  24609  clmvs2  24610  clm0vs  24611  clmopfne  24612  clmvneg1  24615  clmvsneg  24616  clmsubdir  24618  clmvsubval  24625  zlmclm  24628  cmodscmulexp  24638  iscvs  24643  cvsi  24646  isncvsngp  24666  ttgbtwnid  28141  ttgcontlem1  28142
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