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Theorem bnnlm 18978
Description: A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnnlm  |-  ( W  e. Ban  ->  W  e. NrmMod )

Proof of Theorem bnnlm
StepHypRef Expression
1 bnnvc 18977 . 2  |-  ( W  e. Ban  ->  W  e. NrmVec )
2 nvcnlm 18419 . 2  |-  ( W  e. NrmVec  ->  W  e. NrmMod )
31, 2syl 15 1  |-  ( W  e. Ban  ->  W  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1715  NrmModcnlm 18316  NrmVeccnvc 18317  Bancbn 18970
This theorem is referenced by:  bnngp  18979  bnlmod  18980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-nvc 18323  df-bn 18973
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