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Theorem nvclmod 24566
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 24564 . 2 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
2 nlmlmod 24546 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  LModclmod 20704  NrmModcnlm 24440  NrmVeccnvc 24441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-nlm 24446  df-nvc 24447
This theorem is referenced by:  ncvspi  25035  cssbn  25254
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