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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 6103 | . . . 4 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = (Scalar‘𝑊)) | |
2 | islvec.1 | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑓 = 𝑊 → (Scalar‘𝑓) = 𝐹) |
4 | 3 | eleq1d 2672 | . 2 ⊢ (𝑓 = 𝑊 → ((Scalar‘𝑓) ∈ DivRing ↔ 𝐹 ∈ DivRing)) |
5 | df-lvec 18924 | . 2 ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} | |
6 | 4, 5 | elrab2 3333 | 1 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Scalarcsca 15771 DivRingcdr 18570 LModclmod 18686 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-lvec 18924 |
This theorem is referenced by: lvecdrng 18926 lveclmod 18927 lsslvec 18928 lvecprop2d 18987 lvecpropd 18988 rlmlvec 19027 frlmphl 19939 tvclvec 21812 isnvc2 22313 iscvs 22735 cnstrcvs 22749 zclmncvs 22756 lindsdom 32573 lindsenlbs 32574 lduallvec 33459 dvalveclem 35332 dvhlveclem 35415 lmod1zrnlvec 42077 aacllem 42356 |
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