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Theorem nvclmod 22483
 Description: A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nvclmod (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)

Proof of Theorem nvclmod
StepHypRef Expression
1 nvcnlm 22481 . 2 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
2 nlmlmod 22463 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1988  LModclmod 18844  NrmModcnlm 22366  NrmVeccnvc 22367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-nlm 22372  df-nvc 22373 This theorem is referenced by:  ncvspi  22937
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