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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 6907 | . . . 4 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = (Scalar‘𝑊)) | |
2 | islvec.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 1, 2 | eqtr4di 2793 | . . 3 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = 𝐹) |
4 | 3 | eleq1d 2824 | . 2 ⊢ (𝑓 = 𝑊 → ((Scalar‘𝑓) ∈ DivRing ↔ 𝐹 ∈ DivRing)) |
5 | df-lvec 21120 | . 2 ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} | |
6 | 4, 5 | elrab2 3698 | 1 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Scalarcsca 17301 DivRingcdr 20746 LModclmod 20875 LVecclvec 21119 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-lvec 21120 |
This theorem is referenced by: lvecdrng 21122 lveclmod 21123 lsslvec 21126 lmhmlvec 21127 lvecprop2d 21186 lvecpropd 21187 rlmlvec 21229 frlmlvec 21799 frlmphl 21819 mpllvec 22058 tvclvec 24223 isnvc2 24736 iscvs 25174 cnstrcvs 25188 zclmncvs 25196 quslvec 33368 ply1lvec 33565 sralvec 33615 matdim 33643 lmhmlvec2 33647 assalactf1o 33663 ccfldsrarelvec 33696 bj-isvec 37270 lindsdom 37601 lindsenlbs 37602 lduallvec 39136 dvalveclem 41008 dvhlveclem 41091 lmod1zrnlvec 48340 aacllem 49032 |
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