Theorem bnnlm 25481

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 25480	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
2		nvcnlm 24834	. 2 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
3	1, 2	syl 18	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2143  NrmModcnlm 24718  NrmVeccnvc 24719  Bancbn 25473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-nvc 24725  df-bn 25476

This theorem is referenced by:  bnngp  25482  bnlmod  25483