Theorem bnnlm 24603

Description: A Banach space is a normed module. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
bnnlm	⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

Proof of Theorem bnnlm

Step	Hyp	Ref	Expression
1		bnnvc 24602	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
2		nvcnlm 23958	. 2 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2105  NrmModcnlm 23834  NrmVeccnvc 23835  Bancbn 24595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-iota 6425  df-fv 6481  df-nvc 23841  df-bn 24598

This theorem is referenced by:  bnngp  24604  bnlmod  24605