Theorem bnngp 25287

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 25286	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)
2		nlmngp 24620	. 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2114  NrmGrpcngp 24520  NrmModcnlm 24523  Bancbn 25278

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6446  df-fv 6498  df-ov 7361  df-nlm 24529  df-nvc 24530  df-bn 25281

This theorem is referenced by: (None)