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Theorem clmlmod 23663
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 2819 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2819 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 23660 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1139 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cfv 6348  (class class class)co 7148  Basecbs 16475  s cress 16476  Scalarcsca 16560  SubRingcsubrg 19523  LModclmod 19626  fldccnfld 20537  ℂModcclm 23658
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-clm 23659
This theorem is referenced by:  clmgrp  23664  clmabl  23665  clmring  23666  clmfgrp  23667  clmvscl  23684  clmvsass  23685  clmvsdir  23687  clmvsdi  23688  clmvs1  23689  clmvs2  23690  clm0vs  23691  clmopfne  23692  clmvneg1  23695  clmvsneg  23696  clmsubdir  23698  clmvsubval  23705  zlmclm  23708  cmodscmulexp  23718  iscvs  23723  cvsi  23726  isncvsngp  23745  ttgbtwnid  26662  ttgcontlem1  26663
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