Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nvclvec | Structured version Visualization version GIF version |
Description: A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nvclvec | ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvc 22309 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | 1 | simprbi 479 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 LVecclvec 18923 NrmModcnlm 22195 NrmVeccnvc 22196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-nvc 22202 |
This theorem is referenced by: nvctvc 22314 lssnvc 22316 |
Copyright terms: Public domain | W3C validator |