Theorem bnnlm 25537

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 25536	. 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
2		nvcnlm 24890	. 2 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
3	1, 2	syl 18	1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  NrmModcnlm 24774  NrmVeccnvc 24775  Bancbn 25529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-nvc 24781  df-bn 25532

This theorem is used by:  bnngp  25538  bnlmod  25539