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Theorem clmlmod 22675
 Description: A complex 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 2610 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2610 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 22672 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1069 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  Scalarcsca 15771  SubRingcsubrg 18599  LModclmod 18686  ℂfldccnfld 19567  ℂModcclm 22670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-clm 22671 This theorem is referenced by:  clmgrp  22676  clmabl  22677  clmring  22678  clmfgrp  22679  clmvscl  22696  clmvsass  22697  clmvsdir  22699  clmvsdi  22700  clmvs1  22701  clmvs2  22702  clm0vs  22703  clmopfne  22704  clmvneg1  22707  clmvsneg  22708  clmsubdir  22710  clmvsubval  22717  zlmclm  22720  cmodscmulexp  22730  iscvs  22735  cvsi  22738  isncvsngp  22757  ttgbtwnid  25564  ttgcontlem1  25565
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