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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 24522 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 LVecclvec 20935 NrmModcnlm 24399 NrmVeccnvc 24400 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3947 df-nvc 24406 |
This theorem is referenced by: nvclmod 24525 nvctvc 24527 lssnvc 24529 ncvsprp 24990 ncvsm1 24992 ncvsdif 24993 ncvspi 24994 ncvs1 24995 ncvspds 24999 bnnlm 25179 cssbn 25213 |
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