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Theorem bnngp 19157
Description: A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnngp  |-  ( W  e. Ban  ->  W  e. NrmGrp )

Proof of Theorem bnngp
StepHypRef Expression
1 bnnlm 19156 . 2  |-  ( W  e. Ban  ->  W  e. NrmMod )
2 nlmngp 18577 . 2  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
31, 2syl 16 1  |-  ( W  e. Ban  ->  W  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717  NrmGrpcngp 18489  NrmModcnlm 18492  Bancbn 19148
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
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-eu 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016  df-nlm 18498  df-nvc 18499  df-bn 19151
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