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Theorem bnlmod 24507
Description: A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnlmod (𝑊 ∈ Ban → 𝑊 ∈ LMod)

Proof of Theorem bnlmod
StepHypRef Expression
1 bnnlm 24505 . 2 (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2 nlmlmod 23842 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ Ban → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  LModclmod 20123  NrmModcnlm 23736  Bancbn 24497
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-nlm 23742  df-nvc 23743  df-bn 24500
This theorem is referenced by:  sitgclbn  32310
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