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Theorem nlmlmod 22292
Description: A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nlmlmod (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)

Proof of Theorem nlmlmod
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2610 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2610 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2610 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2610 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 22289 . . 3 (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠𝑊)𝑦)) = (((norm‘(Scalar‘𝑊))‘𝑥) · ((norm‘𝑊)‘𝑦))))
87simplbi 475 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
98simp2d 1067 1 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cfv 5804  (class class class)co 6549   · cmul 9820  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  LModclmod 18686  normcnm 22191  NrmGrpcngp 22192  NrmRingcnrg 22194  NrmModcnlm 22195
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-nlm 22201
This theorem is referenced by:  nlmdsdi  22295  nlmdsdir  22296  nlmmul0or  22297  nlmvscnlem2  22299  nlmvscn  22301  nlmtlm  22308  nvclmod  22312  isnvc2  22313  lssnlm  22315  ngpocelbl  22318  idnmhm  22368  0nmhm  22369  nmhmplusg  22371  nmhmcn  22728  cphlmod  22782  bnlmod  22948  nmmulg  29340
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