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Theorem clmlmod 23194
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 2799 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2799 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
31, 2isclm 23191 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)))
43simp1bi 1176 1 (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  Basecbs 16184  s cress 16185  Scalarcsca 16270  SubRingcsubrg 19094  LModclmod 19181  fldccnfld 20068  ℂModcclm 23189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-clm 23190
This theorem is referenced by:  clmgrp  23195  clmabl  23196  clmring  23197  clmfgrp  23198  clmvscl  23215  clmvsass  23216  clmvsdir  23218  clmvsdi  23219  clmvs1  23220  clmvs2  23221  clm0vs  23222  clmopfne  23223  clmvneg1  23226  clmvsneg  23227  clmsubdir  23229  clmvsubval  23236  zlmclm  23239  cmodscmulexp  23249  iscvs  23254  cvsi  23257  isncvsngp  23276  ttgbtwnid  26121  ttgcontlem1  26122
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