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Mirrors > Home > MPE Home > Th. List > isnvc | Structured version Visualization version GIF version |
Description: A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
isnvc | ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nvc 23200 | . 2 ⊢ NrmVec = (NrmMod ∩ LVec) | |
2 | 1 | elin2 4177 | 1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 LVecclvec 19877 NrmModcnlm 23193 NrmVeccnvc 23194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-in 3946 df-nvc 23200 |
This theorem is referenced by: nvcnlm 23308 nvclvec 23309 isnvc2 23311 rlmnvc 23315 isncvsngp 23756 cphnvc 23783 |
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