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Theorem nlmlmod 23214
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 2818 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2818 . . . 4 (norm‘𝑊) = (norm‘𝑊)
3 eqid 2818 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2818 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2818 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6 eqid 2818 . . . 4 (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isnlm 23211 . . 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 498 . 2 (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
98simp2d 1135 1 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145   · cmul 10530  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  LModclmod 19563  normcnm 23113  NrmGrpcngp 23114  NrmRingcnrg 23116  NrmModcnlm 23117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-nlm 23123
This theorem is referenced by:  nlmdsdi  23217  nlmdsdir  23218  nlmmul0or  23219  nlmvscnlem2  23221  nlmvscn  23223  nlmtlm  23230  nvclmod  23234  isnvc2  23235  lssnlm  23237  ngpocelbl  23240  idnmhm  23290  0nmhm  23291  nmhmplusg  23293  nmhmcn  23651  cphlmod  23705  bnlmod  23873  nmmulg  31108
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