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Mirrors > Home > MPE Home > Th. List > islvec | Structured version Visualization version GIF version |
Description: The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.) |
Ref | Expression |
---|---|
islvec.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
islvec | ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . . 4 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = (Scalar‘𝑊)) | |
2 | islvec.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 1, 2 | syl6eqr 2874 | . . 3 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = 𝐹) |
4 | 3 | eleq1d 2897 | . 2 ⊢ (𝑓 = 𝑊 → ((Scalar‘𝑓) ∈ DivRing ↔ 𝐹 ∈ DivRing)) |
5 | df-lvec 19869 | . 2 ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} | |
6 | 4, 5 | elrab2 3682 | 1 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 Scalarcsca 16562 DivRingcdr 19496 LModclmod 19628 LVecclvec 19868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-lvec 19869 |
This theorem is referenced by: lvecdrng 19871 lveclmod 19872 lsslvec 19873 lvecprop2d 19932 lvecpropd 19933 rlmlvec 19972 mpllvec 20227 frlmlvec 20899 frlmphl 20919 tvclvec 22801 isnvc2 23302 iscvs 23725 cnstrcvs 23739 zclmncvs 23746 sralvec 30985 matdim 31008 lmhmlvec2 31012 ccfldsrarelvec 31051 bj-isvec 34563 lindsdom 34880 lindsenlbs 34881 lduallvec 36284 dvalveclem 38155 dvhlveclem 38238 lmhmlvec 39141 lmod1zrnlvec 44543 aacllem 44896 |
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