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Theorem bnngp 24496
Description: A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnngp (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)

Proof of Theorem bnngp
StepHypRef Expression
1 bnnlm 24495 . 2 (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2 nlmngp 23831 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
31, 2syl 17 1 (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  NrmGrpcngp 23723  NrmModcnlm 23726  Bancbn 24487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-nul 5234
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6389  df-fv 6439  df-ov 7272  df-nlm 23732  df-nvc 23733  df-bn 24490
This theorem is referenced by: (None)
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