Theorem bnngp 24850

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

Step	Hyp	Ref	Expression
1		bnnlm 24849	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2		nlmngp 24185	. 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2106  NrmGrpcngp 24077  NrmModcnlm 24080  Bancbn 24841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-nlm 24086  df-nvc 24087  df-bn 24844

This theorem is referenced by: (None)