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Theorem clmlmod 24916
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 2724 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2724 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 24913 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1142 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6533  (class class class)co 7401  Basecbs 17143  s cress 17172  Scalarcsca 17199  SubRingcsubrg 20459  LModclmod 20696  fldccnfld 21228  ℂModcclm 24911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-clm 24912
This theorem is referenced by:  clmgrp  24917  clmabl  24918  clmring  24919  clmfgrp  24920  clmvscl  24937  clmvsass  24938  clmvsdir  24940  clmvsdi  24941  clmvs1  24942  clmvs2  24943  clm0vs  24944  clmopfne  24945  clmvneg1  24948  clmvsneg  24949  clmsubdir  24951  clmvsubval  24958  zlmclm  24961  cmodscmulexp  24971  iscvs  24976  cvsi  24979  isncvsngp  24999  ttgbtwnid  28610  ttgcontlem1  28611
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