Theorem bnnlm 25381

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 25380	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
2		nvcnlm 24734	. 2 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
3	1, 2	syl 17	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2141  NrmModcnlm 24618  NrmVeccnvc 24619  Bancbn 25373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6471  df-fv 6523  df-nvc 24625  df-bn 25376

This theorem is referenced by:  bnngp  25382  bnlmod  25383