| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 6906 | . . . 4 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = (Scalar‘𝑊)) | |
| 2 | islvec.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = 𝐹) |
| 4 | 3 | eleq1d 2826 | . 2 ⊢ (𝑓 = 𝑊 → ((Scalar‘𝑓) ∈ DivRing ↔ 𝐹 ∈ DivRing)) |
| 5 | df-lvec 21102 | . 2 ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} | |
| 6 | 4, 5 | elrab2 3695 | 1 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Scalarcsca 17300 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-lvec 21102 |
| This theorem is referenced by: lvecdrng 21104 lveclmod 21105 lsslvec 21108 lmhmlvec 21109 lvecprop2d 21168 lvecpropd 21169 rlmlvec 21211 frlmlvec 21781 frlmphl 21801 mpllvec 22040 tvclvec 24207 isnvc2 24720 iscvs 25160 cnstrcvs 25174 zclmncvs 25182 quslvec 33388 ply1lvec 33585 sralvec 33636 matdim 33666 lmhmlvec2 33670 assalactf1o 33686 ccfldsrarelvec 33721 fldextrspunlem1 33725 fldextrspunfld 33726 bj-isvec 37288 lindsdom 37621 lindsenlbs 37622 lduallvec 39155 dvalveclem 41027 dvhlveclem 41110 lmod1zrnlvec 48411 aacllem 49320 |
| Copyright terms: Public domain | W3C validator |