![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nvcnlm | Structured version Visualization version GIF version |
Description: A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nvcnlm | ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvc 23301 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 LVecclvec 19867 NrmModcnlm 23187 NrmVeccnvc 23188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-nvc 23194 |
This theorem is referenced by: nvclmod 23304 nvctvc 23306 lssnvc 23308 ncvsprp 23757 ncvsm1 23759 ncvsdif 23760 ncvspi 23761 ncvs1 23762 ncvspds 23766 bnnlm 23945 cssbn 23979 |
Copyright terms: Public domain | W3C validator |