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

Step	Hyp	Ref	Expression
1		bnnlm 24495	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2		nlmngp 23831	. 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)

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)