MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bnnlm Unicode version

Theorem bnnlm 19255
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 19254 . 2  |-  ( W  e. Ban  ->  W  e. NrmVec )
2 nvcnlm 18692 . 2  |-  ( W  e. NrmVec  ->  W  e. NrmMod )
31, 2syl 16 1  |-  ( W  e. Ban  ->  W  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721  NrmModcnlm 18589  NrmVeccnvc 18590  Bancbn 19247
This theorem is referenced by:  bnngp  19256  bnlmod  19257  sitgclbn  24618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-nvc 18596  df-bn 19250
  Copyright terms: Public domain W3C validator