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Mirrors > Home > MPE Home > Th. List > tvclvec | Structured version Visualization version GIF version |
Description: A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
tvclvec | ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tvclmod 22806 | . 2 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod) | |
2 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | 2 | tvctdrg 22801 | . . 3 ⊢ (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ TopDRing) |
4 | tdrgdrng 22782 | . . 3 ⊢ ((Scalar‘𝑊) ∈ TopDRing → (Scalar‘𝑊) ∈ DivRing) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝑊 ∈ TopVec → (Scalar‘𝑊) ∈ DivRing) |
6 | 2 | islvec 19876 | . 2 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ DivRing)) |
7 | 1, 5, 6 | sylanbrc 585 | 1 ⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6355 Scalarcsca 16568 DivRingcdr 19502 LModclmod 19634 LVecclvec 19874 TopDRingctdrg 22765 TopVecctvc 22767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-lvec 19875 df-tdrg 22769 df-tlm 22770 df-tvc 22771 |
This theorem is referenced by: (None) |
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