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Mirrors > Home > MPE Home > Th. List > lvecdrng | Structured version Visualization version GIF version |
Description: The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.) |
Ref | Expression |
---|---|
islvec.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
lvecdrng | ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islvec.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | islvec 20366 | . 2 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
3 | 2 | simprbi 497 | 1 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 Scalarcsca 16965 DivRingcdr 19991 LModclmod 20123 LVecclvec 20364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-lvec 20365 |
This theorem is referenced by: lsslvec 20369 lvecvs0or 20370 lssvs0or 20372 lvecinv 20375 lspsnvs 20376 lspsneq 20384 lspfixed 20390 lspexch 20391 lspsolv 20405 islbs2 20416 islbs3 20417 obsne0 20932 islinds4 21042 nvctvc 23864 lssnvc 23866 cvsunit 24294 cvsdivcl 24296 cphsubrg 24344 cphreccl 24345 cphqss 24352 phclm 24396 ipcau2 24398 tcphcph 24401 hlprlem 24531 ishl2 24534 0nellinds 31566 lmhmlvec2 31702 lfl1 37084 lkrsc 37111 eqlkr3 37115 lkrlsp 37116 lkrshp 37119 lduallvec 37168 dochkr1 39492 dochkr1OLDN 39493 lcfl7lem 39513 lclkrlem2m 39533 lclkrlem2o 39535 lclkrlem2p 39536 lcfrlem1 39556 lcfrlem2 39557 lcfrlem3 39558 lcfrlem29 39585 lcfrlem31 39587 lcfrlem33 39589 mapdpglem17N 39702 mapdpglem18 39703 mapdpglem19 39704 mapdpglem21 39706 mapdpglem22 39707 hdmapip1 39930 hgmapvvlem1 39937 hgmapvvlem2 39938 hgmapvvlem3 39939 prjspersym 40446 lincreslvec3 45823 isldepslvec2 45826 |
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