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Theorem bnngp 19285
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 19284 . 2  |-  ( W  e. Ban  ->  W  e. NrmMod )
2 nlmngp 18703 . 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 1725  NrmGrpcngp 18615  NrmModcnlm 18618  Bancbn 19276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlm 18624  df-nvc 18625  df-bn 19279
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