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Theorem List for Metamath Proof Explorer - 33201-33300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremlsatelbN 33201 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 33202 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 33203* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 33999 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 33204* Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 18853. (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 33205 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 27683 analog.) Explicit atom version of lsmcv 18866. (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 33206* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 28389 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈 ≠ { 0 })       (𝜑 → ∃𝑞𝐴 𝑞𝑈)
 
Theoremlssats 33207* 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 28392 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 = (𝑁 {𝑥𝐴𝑥𝑈}))
 
Theoremlpssat 33208* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 28394 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝑈)       (𝜑 → ∃𝑞𝐴 (𝑞𝑈 ∧ ¬ 𝑞𝑇))
 
Theoremlrelat 33209* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 28395 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 33210* The ordering of two subspaces is determined by the atoms under them. (chrelat3 28402 analog.) (Contributed by NM, 29-Oct-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝑈 ↔ ∀𝑝𝐴 (𝑝𝑇𝑝𝑈)))
 
Theoremlssat 33211* 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 28394 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆𝑉𝑆) ∧ 𝑈𝑉) → ∃𝑝𝐴 (𝑝𝑉 ∧ ¬ 𝑝𝑈))
 
Theoremislshpat 33212* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 33175. (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 33213 Extend class notation with the covering relation for a left module or left vector space.
class L
 
Definitiondf-lcv 33214* 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 33216 for binary relation. (df-cv 28310 analog.) (Contributed by NM, 7-Jan-2015.)
L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
 
Theoremlcvfbr 33215* 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 33216* The covers relation for a left vector space (or a left module). (cvbr 28313 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
 
Theoremlcvbr2 33217* The covers relation for a left vector space (or a left module). (cvbr2 28314 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
 
Theoremlcvbr3 33218* 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 33219 The covers relation implies proper subset. (cvpss 28316 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶𝑈)       (𝜑𝑇𝑈)
 
Theoremlcvnbtwn 33220 The covers relation implies no in-betweenness. (cvnbtwn 28317 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)       (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
 
Theoremlcvntr 33221 The covers relation is not transitive. (cvntr 28323 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑇𝐶𝑈)       (𝜑 → ¬ 𝑅𝐶𝑈)
 
Theoremlcvnbtwn2 33222 The covers relation implies no in-betweenness. (cvnbtwn2 28318 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑇)
 
Theoremlcvnbtwn3 33223 The covers relation implies no in-betweenness. (cvnbtwn3 28319 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑅)
 
Theoremlsmcv2 33224 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 28324 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 33225* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 28397 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 33226 An atom covers the zero subspace. (atcv0 28373 analog.) (Contributed by NM, 7-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)       (𝜑 → { 0 }𝐶𝑄)
 
Theoremlsatcveq0 33227 A subspace covered by an atom must be the zero subspace. (atcveq0 28379 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 33228 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 33229 Lemma for lcvexch 33234. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇 ⊊ (𝑇 𝑈) ↔ (𝑇𝑈) ⊊ 𝑈))
 
Theoremlcvexchlem2 33230 Lemma for lcvexch 33234. (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 33231 Lemma for lcvexch 33234. (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 33232 Lemma for lcvexch 33234. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶(𝑇 𝑈))       (𝜑 → (𝑇𝑈)𝐶𝑈)
 
Theoremlcvexchlem5 33233 Lemma for lcvexch 33234. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑 → (𝑇𝑈)𝐶𝑈)       (𝜑𝑇𝐶(𝑇 𝑈))
 
Theoremlcvexch 33234 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 28400 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 33235 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 28406 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 33236 Covering property of a subspace plus an atom. (chcv1 28386 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 33237 Covering property of a subspace plus an atom. (chcv2 28387 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 33238 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 28412 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 33239 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 28407 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (¬ 𝑄𝑈 ↔ (𝑈𝑄) = { 0 }))
 
Theoremlsatnem0 33240 The meet of distinct atoms is the zero subspace. (atnemeq0 28408 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)       (𝜑 → (𝑄𝑅 ↔ (𝑄𝑅) = { 0 }))
 
Theoremlsatexch1 33241 The atom exch1ange property. (hlatexch1 33589 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 33242 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 28410 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 33243 Two atoms covering the zero subspace are equal. (atcv1 28411 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 33244 Lemma for lsatcvat 33245. (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 33245 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 28417 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 33246 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 28418 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 33247 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 28427 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 33248 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 33249 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 33669 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 33250 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 33670 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 33251 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 34237 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 33249 and l1cvat 33250 to 𝑈𝐻, which in turn change 𝑈𝐻 in islshpcv 33248 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)
 
20.22.7  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 33252 Extend class notation with all linear functionals of a left module or left vector space.
class LFnl
 
Definitiondf-lfl 33253* 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 33254* 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 33255* 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 33256 Property of a linear functional. (lnfnli 28071 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 33257* Properties that determine a linear functional. TODO: use this in place of islfl 33255 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 33258 A linear functional is a function from vectors to scalars. (lnfnfi 28072 analog.) (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑋𝐺𝐹) → 𝐺:𝑉𝐾)
 
Theoremlflcl 33259 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 33260 A linear functional is zero at the zero vector. (lnfn0i 28073 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 33261 Property of a linear functional. (lnfnaddi 28074 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 33262 Property of a linear functional. (lnfnaddi 28074 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 33263 Property of a linear functional. (lnfnmuli 28075 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 33264 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 33265* 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 33266 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)
 
Theoremlfladdcom 33267 Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)       (𝜑 → (𝐺𝑓 + 𝐻) = (𝐻𝑓 + 𝐺))
 
Theoremlfladdass 33268 Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
𝑅 = (Scalar‘𝑊)    &    + = (+g𝑅)    &   𝐹 = (LFnl‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑𝐼𝐹)       (𝜑 → ((𝐺𝑓 + 𝐻) ∘𝑓 + 𝐼) = (𝐺𝑓 + (𝐻𝑓 + 𝐼)))
 
Theoremlfladd0l 33269 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 33270* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 33341, 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 33271* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 33341, 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 33272 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 33273 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 33274 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 33275 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 33276 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 33277 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 33278 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 33279 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 33280 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 33281* 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 33282* 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 33283 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 33284 Membership in the kernel of a functional. (elnlfn 27959 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 33285* 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 33286 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 33287 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 33288 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 33289 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 33290 The kernel of a linear functional is a subspace. (nlelshi 28091 analog.) (Contributed by NM, 16-Apr-2014.)
𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐺𝐹) → (𝐾𝐺) ∈ 𝑆)
 
Theoremlkrssv 33291 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺𝐹)       (𝜑 → (𝐾𝐺) ⊆ 𝑉)
 
Theoremlkrsc 33292 The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑅𝐾)    &    0 = (0g𝐷)    &   (𝜑𝑅0 )       (𝜑 → (𝐿‘(𝐺𝑓 · (𝑉 × {𝑅}))) = (𝐿𝐺))
 
Theoremlkrscss 33293 The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐺𝐹)    &   (𝜑𝑅𝐾)       (𝜑 → (𝐿𝐺) ⊆ (𝐿‘(𝐺𝑓 · (𝑉 × {𝑅}))))
 
Theoremeqlkr 33294* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝐺𝐹𝐻𝐹) ∧ (𝐿𝐺) = (𝐿𝐻)) → ∃𝑟𝐾𝑥𝑉 (𝐻𝑥) = ((𝐺𝑥) · 𝑟))
 
Theoremeqlkr2 33295* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &    · = (.r𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐿 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝐺𝐹𝐻𝐹) ∧ (𝐿𝐺) = (𝐿𝐻)) → ∃𝑟𝐾 𝐻 = (𝐺𝑓 · (𝑉 × {𝑟})))
 
Theoremeqlkr3 33296 Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝐺𝐹)    &   (𝜑𝐻𝐹)    &   (𝜑 → (𝐾𝐺) = (𝐾𝐻))    &   (𝜑 → (𝐺𝑋) = (𝐻𝑋))    &   (𝜑 → (𝐺𝑋) ≠ 0 )       (𝜑𝐺 = 𝐻)
 
Theoremlkrlsp 33297 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 33284) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑋𝑉𝐺𝐹) ∧ (𝐺𝑋) ≠ 0 ) → ((𝐾𝐺) (𝑁‘{𝑋})) = 𝑉)
 
Theoremlkrlsp2 33298 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑋𝑉𝐺𝐹) ∧ ¬ 𝑋 ∈ (𝐾𝐺)) → ((𝐾𝐺) (𝑁‘{𝑋})) = 𝑉)
 
Theoremlkrlsp3 33299 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑋𝑉𝐺𝐹) ∧ ¬ 𝑋 ∈ (𝐾𝐺)) → (𝑁‘((𝐾𝐺) ∪ {𝑋})) = 𝑉)
 
Theoremlkrshp 33300 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐷 = (Scalar‘𝑊)    &    0 = (0g𝐷)    &   𝐻 = (LSHyp‘𝑊)    &   𝐹 = (LFnl‘𝑊)    &   𝐾 = (LKer‘𝑊)       ((𝑊 ∈ LVec ∧ 𝐺𝐹𝐺 ≠ (𝑉 × { 0 })) → (𝐾𝐺) ∈ 𝐻)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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