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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 23231 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 LVecclvec 19803 NrmModcnlm 23117 NrmVeccnvc 23118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-nvc 23124 |
This theorem is referenced by: nvclmod 23234 nvctvc 23236 lssnvc 23238 ncvsprp 23683 ncvsm1 23685 ncvsdif 23686 ncvspi 23687 ncvs1 23688 ncvspds 23692 bnnlm 23871 cssbn 23905 |
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