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Theorem bnlmod 24108
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 24106 . 2 (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2 nlmlmod 23444 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ Ban → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  LModclmod 19766  NrmModcnlm 23346  Bancbn 24098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6308  df-fv 6358  df-ov 7186  df-nlm 23352  df-nvc 23353  df-bn 24101
This theorem is referenced by:  sitgclbn  31893
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