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Theorem bnlmod 23549
 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 23547 . 2 (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2 nlmlmod 22890 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ Ban → 𝑊 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107  LModclmod 19255  NrmModcnlm 22793  Bancbn 23539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-nlm 22799  df-nvc 22800  df-bn 23542 This theorem is referenced by:  sitgclbn  31003
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