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Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsatlss 33301 The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊 ∈ LMod → 𝐴𝑆)

Theoremlsatlssel 33302 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈𝑆)

Theoremlsatssv 33303 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)       (𝜑𝑄𝑉)

Theoremlsatn0 33304 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 28588 analog.) (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈 ≠ { 0 })

Theoremlsatspn0 33305 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴𝑋0 ))

Theoremlsator0sp 33306 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ∨ (𝑁‘{𝑋}) = { 0 }))

Theoremlsatssn0 33307 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)    &   (𝜑𝑄𝑈)       (𝜑𝑈 ≠ { 0 })

Theoremlsatcmp 33308 If two atoms are comparable, they are equal. (atsseq 28590 analog.) TODO: can lspsncmp 18937 shorten this? (Contributed by NM, 25-Aug-2014.)
𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑𝑈𝐴)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

Theoremlsatcmp2 33309 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 33308. TODO: can lspsncmp 18937 shorten this? (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑 → (𝑈𝐴𝑈 = { 0 }))       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

Theoremlsatel 33310 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐴)    &   (𝜑𝑋𝑈)    &   (𝜑𝑋0 )       (𝜑𝑈 = (𝑁‘{𝑋}))

TheoremlsatelbN 33311 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑈𝐴)       (𝜑 → (𝑋𝑈𝑈 = (𝑁‘{𝑋})))

Theoremlsat2el 33312 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)    &   (𝜑𝑋0 )    &   (𝜑𝑋𝑃)    &   (𝜑𝑋𝑄)       (𝜑𝑃 = 𝑄)

Theoremlsmsat 33313* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 34109 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑇 ≠ { 0 })    &   (𝜑𝑄 ⊆ (𝑇 𝑈))       (𝜑 → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))

TheoremlsatfixedN 33314* Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 18949. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑄 ≠ (𝑁‘{𝑋}))    &   (𝜑𝑄 ≠ (𝑁‘{𝑌}))    &   (𝜑𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))

Theoremlsmsatcv 33315 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 27895 analog.) Explicit atom version of lsmcv 18962. (Contributed by NM, 29-Oct-2014.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       ((𝜑𝑇𝑈𝑈 ⊆ (𝑇 𝑄)) → 𝑈 = (𝑇 𝑄))

Theoremlssatomic 33316* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 28601 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈 ≠ { 0 })       (𝜑 → ∃𝑞𝐴 𝑞𝑈)

Theoremlssats 33317* The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 28604 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 = (𝑁 {𝑥𝐴𝑥𝑈}))

Theoremlpssat 33318* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 28606 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝑈)       (𝜑 → ∃𝑞𝐴 (𝑞𝑈 ∧ ¬ 𝑞𝑇))

Theoremlrelat 33319* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 28607 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝑈)       (𝜑 → ∃𝑞𝐴 (𝑇 ⊊ (𝑇 𝑞) ∧ (𝑇 𝑞) ⊆ 𝑈))

Theoremlssatle 33320* The ordering of two subspaces is determined by the atoms under them. (chrelat3 28614 analog.) (Contributed by NM, 29-Oct-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝑈 ↔ ∀𝑝𝐴 (𝑝𝑇𝑝𝑈)))

Theoremlssat 33321* Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 28606 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆𝑉𝑆) ∧ 𝑈𝑉) → ∃𝑝𝐴 (𝑝𝑉 ∧ ¬ 𝑝𝑈))

Theoremislshpat 33322* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 33285. (Contributed by NM, 11-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑞𝐴 (𝑈 𝑞) = 𝑉)))

Syntaxclcv 33323 Extend class notation with the covering relation for a left module or left vector space.
class L

Definitiondf-lcv 33324* Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation 𝐴( ⋖L𝑊)𝐵 is read "𝐵 covers 𝐴 " or "𝐴 is covered by 𝐵 " , and it means that 𝐵 is larger than 𝐴 and there is nothing in between. See lcvbr 33326 for binary relation. (df-cv 28522 analog.) (Contributed by NM, 7-Jan-2015.)
L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})

Theoremlcvfbr 33325* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)       (𝜑𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡𝑆𝑢𝑆) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠𝑆 (𝑡𝑠𝑠𝑢)))})

Theoremlcvbr 33326* The covers relation for a left vector space (or a left module). (cvbr 28525 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))

Theoremlcvbr2 33327* The covers relation for a left vector space (or a left module). (cvbr2 28526 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))

Theoremlcvbr3 33328* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → (𝑠 = 𝑇𝑠 = 𝑈)))))

Theoremlcvpss 33329 The covers relation implies proper subset. (cvpss 28528 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶𝑈)       (𝜑𝑇𝑈)

Theoremlcvnbtwn 33330 The covers relation implies no in-betweenness. (cvnbtwn 28529 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)       (𝜑 → ¬ (𝑅𝑈𝑈𝑇))

Theoremlcvntr 33331 The covers relation is not transitive. (cvntr 28535 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑇𝐶𝑈)       (𝜑 → ¬ 𝑅𝐶𝑈)

Theoremlcvnbtwn2 33332 The covers relation implies no in-betweenness. (cvnbtwn2 28530 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑇)

Theoremlcvnbtwn3 33333 The covers relation implies no in-betweenness. (cvnbtwn3 28531 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑅)

Theoremlsmcv2 33334 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 28536 analog.) (Contributed by NM, 10-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ 𝑈)       (𝜑𝑈𝐶(𝑈 (𝑁‘{𝑋})))

Theoremlcvat 33335* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 28609 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶𝑈)       (𝜑 → ∃𝑞𝐴 (𝑇 𝑞) = 𝑈)

Theoremlsatcv0 33336 An atom covers the zero subspace. (atcv0 28585 analog.) (Contributed by NM, 7-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)       (𝜑 → { 0 }𝐶𝑄)

Theoremlsatcveq0 33337 A subspace covered by an atom must be the zero subspace. (atcveq0 28591 analog.) (Contributed by NM, 7-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑈𝐶𝑄𝑈 = { 0 }))

Theoremlsat0cv 33338 A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑈𝐴 ↔ { 0 }𝐶𝑈))

Theoremlcvexchlem1 33339 Lemma for lcvexch 33344. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇 ⊊ (𝑇 𝑈) ↔ (𝑇𝑈) ⊊ 𝑈))

Theoremlcvexchlem2 33340 Lemma for lcvexch 33344. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝑆)    &   (𝜑 → (𝑇𝑈) ⊆ 𝑅)    &   (𝜑𝑅𝑈)       (𝜑 → ((𝑅 𝑇) ∩ 𝑈) = 𝑅)

Theoremlcvexchlem3 33341 Lemma for lcvexch 33344. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑅)    &   (𝜑𝑅 ⊆ (𝑇 𝑈))       (𝜑 → ((𝑅𝑈) 𝑇) = 𝑅)

Theoremlcvexchlem4 33342 Lemma for lcvexch 33344. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶(𝑇 𝑈))       (𝜑 → (𝑇𝑈)𝐶𝑈)

Theoremlcvexchlem5 33343 Lemma for lcvexch 33344. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑 → (𝑇𝑈)𝐶𝑈)       (𝜑𝑇𝐶(𝑇 𝑈))

Theoremlcvexch 33344 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 28612 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → ((𝑇𝑈)𝐶𝑈𝑇𝐶(𝑇 𝑈)))

Theoremlcvp 33345 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 28618 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → ((𝑈𝑄) = { 0 } ↔ 𝑈𝐶(𝑈 𝑄)))

Theoremlcv1 33346 Covering property of a subspace plus an atom. (chcv1 28598 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (¬ 𝑄𝑈𝑈𝐶(𝑈 𝑄)))

Theoremlcv2 33347 Covering property of a subspace plus an atom. (chcv2 28599 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑈 ⊊ (𝑈 𝑄) ↔ 𝑈𝐶(𝑈 𝑄)))

Theoremlsatexch 33348 The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 28624 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑄 ⊆ (𝑈 𝑅))    &   (𝜑 → (𝑈𝑄) = { 0 })       (𝜑𝑅 ⊆ (𝑈 𝑄))

Theoremlsatnle 33349 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 28619 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (¬ 𝑄𝑈 ↔ (𝑈𝑄) = { 0 }))

Theoremlsatnem0 33350 The meet of distinct atoms is the zero subspace. (atnemeq0 28620 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)       (𝜑 → (𝑄𝑅 ↔ (𝑄𝑅) = { 0 }))

Theoremlsatexch1 33351 The atom exch1ange property. (hlatexch1 33699 analog.) (Contributed by NM, 14-Jan-2015.)
= (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑆𝐴)    &   (𝜑𝑄 ⊆ (𝑆 𝑅))    &   (𝜑𝑄𝑆)       (𝜑𝑅 ⊆ (𝑆 𝑄))

Theoremlsatcv0eq 33352 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 28622 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)       (𝜑 → ({ 0 }𝐶(𝑄 𝑅) ↔ 𝑄 = 𝑅))

Theoremlsatcv1 33353 Two atoms covering the zero subspace are equal. (atcv1 28623 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑈𝐶(𝑄 𝑅))       (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅))

Theoremlsatcvatlem 33354 Lemma for lsatcvat 33355. (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑈 ≠ { 0 })    &   (𝜑𝑈 ⊊ (𝑄 𝑅))    &   (𝜑 → ¬ 𝑄𝑈)       (𝜑𝑈𝐴)

Theoremlsatcvat 33355 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 28629 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑈 ≠ { 0 })    &   (𝜑𝑈 ⊊ (𝑄 𝑅))       (𝜑𝑈𝐴)

Theoremlsatcvat2 33356 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 28630 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑄𝑅)    &   (𝜑𝑈𝐶(𝑄 𝑅))       (𝜑𝑈𝐴)

Theoremlsatcvat3 33357 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 28639 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑄𝑅)    &   (𝜑 → ¬ 𝑅𝑈)    &   (𝜑𝑄 ⊆ (𝑈 𝑅))       (𝜑 → (𝑈 ∩ (𝑄 𝑅)) ∈ 𝐴)

Theoremislshpcv 33358 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)       (𝜑 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝐶𝑉)))

Theoreml1cvpat 33359 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 33779 analog.) (Contributed by NM, 11-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑈𝐶𝑉)    &   (𝜑 → ¬ 𝑄𝑈)       (𝜑 → (𝑈 𝑄) = 𝑉)

Theoreml1cvat 33360 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 33780 analog.) (Contributed by NM, 11-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑄𝑅)    &   (𝜑𝑈𝐶𝑉)    &   (𝜑 → ¬ 𝑄𝑈)       (𝜑 → ((𝑄 𝑅) ∩ 𝑈) ∈ 𝐴)

Theoremlshpat 33361 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 34347 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 33359 and l1cvat 33360 to 𝑈𝐻, which in turn change 𝑈𝐻 in islshpcv 33358 to 𝑈𝐶𝑉, with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)    &   (𝜑𝑄𝑅)    &   (𝜑 → ¬ 𝑄𝑈)       (𝜑 → ((𝑄 𝑅) ∩ 𝑈) ∈ 𝐴)

21.22.7  Functionals and kernels of a left vector space (or module)

Syntaxclfn 33362 Extend class notation with all linear functionals of a left module or left vector space.
class LFnl

Definitiondf-lfl 33363* Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑𝑚 (Base‘𝑤)) ∣ ∀𝑟 ∈ (Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠𝑤)𝑥)(+g𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓𝑥))(+g‘(Scalar‘𝑤))(𝑓𝑦))})

Theoremlflset 33364* The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐷 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐷)    &    = (+g𝐷)    &    × = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)       (𝑊𝑋𝐹 = {𝑓 ∈ (𝐾𝑚 𝑉) ∣ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓𝑥)) (𝑓𝑦))})

Theoremislfl 33365* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐷 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐷)    &    = (+g𝐷)    &    × = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)       (𝑊𝑋 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))

Theoremlfli 33366 Property of a linear functional. (lnfnli 28283 analog.) (Contributed by NM, 16-Apr-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐷 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐷)    &    = (+g𝐷)    &    × = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))

Theoremislfld 33367* Properties that determine a linear functional. TODO: use this in place of islfl 33365 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑𝐷 = (Scalar‘𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝐾 = (Base‘𝐷))    &   (𝜑 = (+g𝐷))    &   (𝜑× = (.r𝐷))    &   (𝜑𝐹 = (LFnl‘𝑊))    &   (𝜑𝐺:𝑉𝐾)    &   ((𝜑 ∧ (𝑟𝐾𝑥𝑉𝑦𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))    &   (𝜑𝑊𝑋)       (𝜑𝐺𝐹)

Theoremlflf 33368 A linear functional is a function from vectors to scalars. (lnfnfi 28284 analog.) (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑋𝐺𝐹) → 𝐺:𝑉𝐾)

Theoremlflcl 33369 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑌𝐺𝐹𝑋𝑉) → (𝐺𝑋) ∈ 𝐾)

Theoremlfl0 33370 A linear functional is zero at the zero vector. (lnfn0i 28285 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑍 = (0g𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐺𝑍) = 0 )

Theoremlfladd 33371 Property of a linear functional. (lnfnaddi 28286 analog.) (Contributed by NM, 18-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    = (+g𝐷)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑋𝑉𝑌𝑉)) → (𝐺‘(𝑋 + 𝑌)) = ((𝐺𝑋) (𝐺𝑌)))

Theoremlflsub 33372 Property of a linear functional. (lnfnaddi 28286 analog.) (Contributed by NM, 18-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝑀 = (-g𝐷)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑋𝑉𝑌𝑉)) → (𝐺‘(𝑋 𝑌)) = ((𝐺𝑋)𝑀(𝐺𝑌)))

Theoremlflmul 33373 Property of a linear functional. (lnfnmuli 28287 analog.) (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    × = (.r𝐷)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉)) → (𝐺‘(𝑅 · 𝑋)) = (𝑅 × (𝐺𝑋)))

Theoremlfl0f 33374 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ 𝐹)

Theoremlfl1 33375* A nonzero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &    1 = (1r𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊 ∈ LVec ∧ 𝐺𝐹𝐺 ≠ (𝑉 × { 0 })) → ∃𝑥𝑉 (𝐺𝑥) = 1 )

Theoremlfladdcl 33376 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺𝑓 + 𝐻) ∈ 𝐹)

𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺𝑓 + 𝐻) = (𝐻𝑓 + 𝐺))

𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑𝐼𝐹)       (𝜑 → ((𝐺𝑓 + 𝐻) ∘𝑓 + 𝐼) = (𝐺𝑓 + (𝐻𝑓 + 𝐼)))

Theoremlfladd0l 33379 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &    0 = (0g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑉 × { 0 }) ∘𝑓 + 𝐺) = 𝐺)

Theoremlflnegcl 33380* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 33451, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐼 = (invg𝑅)    &   𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥)))    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑𝑁𝐹)

Theoremlflnegl 33381* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 33451, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐼 = (invg𝑅)    &   𝑁 = (𝑥𝑉 ↦ (𝐼‘(𝐺𝑥)))    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &    + = (+g𝑅)    &    0 = (0g𝑅)       (𝜑 → (𝑁𝑓 + 𝐺) = (𝑉 × { 0 }))

Theoremlflvscl 33382 Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝑅𝐾)       (𝜑 → (𝐺𝑓 · (𝑉 × {𝑅})) ∈ 𝐹)

Theoremlflvsdi1 33383 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → ((𝐺𝑓 + 𝐻) ∘𝑓 · (𝑉 × {𝑋})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐻𝑓 · (𝑉 × {𝑋}))))

Theoremlflvsdi2 33384 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · ((𝑉 × {𝑋}) ∘𝑓 + (𝑉 × {𝑌}))) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))

Theoremlflvsdi2a 33385 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 + (𝐺𝑓 · (𝑉 × {𝑌}))))

Theoremlflvsass 33386 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐾)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × {(𝑋 · 𝑌)})) = ((𝐺𝑓 · (𝑉 × {𝑋})) ∘𝑓 · (𝑉 × {𝑌})))

Theoremlfl0sc 33387 The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of (𝑉 × { 0 }) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × { 0 })) = (𝑉 × { 0 }))

Theoremlflsc0N 33388 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &    0 = (0g𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐾)       (𝜑 → ((𝑉 × { 0 }) ∘𝑓 · (𝑉 × {𝑋})) = (𝑉 × { 0 }))

Theoremlfl1sc 33389 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &    1 = (1r𝐷)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝑓 · (𝑉 × { 1 })) = 𝐺)

Syntaxclk 33390 Extend class notation with the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space.
class LKer

Definitiondf-lkr 33391* Define the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))

Theoremlkrfval 33392* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))

Theoremlkrval 33393 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))

Theoremellkr 33394 Membership in the kernel of a functional. (elnlfn 28171 analog.) (Contributed by NM, 16-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))

Theoremlkrval2 33395* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })

Theoremellkr2 33396 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊𝑌)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))

Theoremlkrcl 33397 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑌𝐺𝐹𝑋 ∈ (𝐾𝐺)) → 𝑋𝑉)

Theoremlkrf0 33398 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊𝑌𝐺𝐹𝑋 ∈ (𝐾𝐺)) → (𝐺𝑋) = 0 )

Theoremlkr0f 33399 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → ((𝐾𝐺) = 𝑉𝐺 = (𝑉 × { 0 })))

Theoremlkrlss 33400 The kernel of a linear functional is a subspace. (nlelshi 28303 analog.) (Contributed by NM, 16-Apr-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐾𝐺) ∈ 𝑆)

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