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Theorem bnnlm 19299
 Description: A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnnlm Ban NrmMod

Proof of Theorem bnnlm
StepHypRef Expression
1 bnnvc 19298 . 2 Ban NrmVec
2 nvcnlm 18736 . 2 NrmVec NrmMod
31, 2syl 16 1 Ban NrmMod
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1726  NrmModcnlm 18633  NrmVeccnvc 18634  Bancbn 19291 This theorem is referenced by:  bnngp  19300  bnlmod  19301  sitgclbn  24662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-nvc 18640  df-bn 19294
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