isnvc - Metamath Proof Explorer

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Theorem isnvc 24625

Description: A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)

**Assertion**
Ref	Expression
isnvc	⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

**Proof of Theorem isnvc**
Step	Hyp	Ref	Expression
1		df-nvc 24509	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4192	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2098 LVecclvec 20986 NrmModcnlm 24502 NrmVeccnvc 24503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696

This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3465 df-in 3948 df-nvc 24509

This theorem is referenced by: nvcnlm 24626 nvclvec 24627 isnvc2 24629 rlmnvc 24633 isncvsngp 25090 cphnvc 25117

	
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