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Theorem clmlmod 25100
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 2737 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2737 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 25097 . 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 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Scalarcsca 17300  SubRingcsubrg 20569  LModclmod 20858  fldccnfld 21364  ℂModcclm 25095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-clm 25096
This theorem is referenced by:  clmgrp  25101  clmabl  25102  clmring  25103  clmfgrp  25104  clmvscl  25121  clmvsass  25122  clmvsdir  25124  clmvsdi  25125  clmvs1  25126  clmvs2  25127  clm0vs  25128  clmopfne  25129  clmvneg1  25132  clmvsneg  25133  clmsubdir  25135  clmvsubval  25142  zlmclm  25145  cmodscmulexp  25155  iscvs  25160  cvsi  25163  isncvsngp  25183  ttgbtwnid  28898  ttgcontlem1  28899
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