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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 22997 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | 1 | simplbi 490 | 1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2048 LVecclvec 19586 NrmModcnlm 22883 NrmVeccnvc 22884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-in 3832 df-nvc 22890 |
This theorem is referenced by: nvclmod 23000 nvctvc 23002 lssnvc 23004 ncvsprp 23449 ncvsm1 23451 ncvsdif 23452 ncvspi 23453 ncvs1 23454 ncvspds 23458 bnnlm 23637 cssbn 23671 |
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