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Theorem bnlmod 25291
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 25289 . 2 (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2 nlmlmod 24615 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ Ban → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  LModclmod 20750  NrmModcnlm 24509  Bancbn 25281
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 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-nlm 24515  df-nvc 24516  df-bn 25284
This theorem is referenced by:  sitgclbn  33996
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