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Theorem nvclmod 24085
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 24083 . 2 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
2 nlmlmod 24065 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  LModclmod 20365  NrmModcnlm 23959  NrmVeccnvc 23960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-nlm 23965  df-nvc 23966
This theorem is referenced by:  ncvspi  24543  cssbn  24762
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