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Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlshpnel 36001 A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑 → ¬ 𝑋𝑈)
 
Theoremlshpnelb 36002 The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)       (𝜑 → (¬ 𝑋𝑈 ↔ (𝑈 (𝑁‘{𝑋})) = 𝑉))
 
Theoremlshpnel2N 36003 Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑 → (𝑈𝐻 ↔ (𝑈 (𝑁‘{𝑋})) = 𝑉))
 
Theoremlshpne0 36004 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.) (Proof shortened by AV, 19-Jul-2022.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑𝑋0 )
 
Theoremlshpdisj 36005 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝐻)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝑈 (𝑁‘{𝑋})) = 𝑉)       (𝜑 → (𝑈 ∩ (𝑁‘{𝑋})) = { 0 })
 
Theoremlshpcmp 36006 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐻)    &   (𝜑𝑈𝐻)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))
 
TheoremlshpinN 36007 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐻)    &   (𝜑𝑈𝐻)       (𝜑 → ((𝑇𝑈) ∈ 𝐻𝑇 = 𝑈))
 
Theoremlsatset 36008* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊𝑋𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
 
Theoremislsat 36009* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (𝑊𝑋 → (𝑈𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})))
 
Theoremlsatlspsn2 36010 The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 36011 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑋0 ) → (𝑁‘{𝑋}) ∈ 𝐴)
 
Theoremlsatlspsn 36011 The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
 
Theoremislsati 36012* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊𝑋𝑈𝐴) → ∃𝑣𝑉 𝑈 = (𝑁‘{𝑣}))
 
Theoremlsateln0 36013* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑 → ∃𝑣𝑈 𝑣0 )
 
Theoremlsatlss 36014 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 36015 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈𝑆)
 
Theoremlsatssv 36016 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)       (𝜑𝑄𝑉)
 
Theoremlsatn0 36017 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 30050 analog.) (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈 ≠ { 0 })
 
Theoremlsatspn0 36018 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 36019 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 36020 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 36021 If two atoms are comparable, they are equal. (atsseq 30052 analog.) TODO: can lspsncmp 19819 shorten this? (Contributed by NM, 25-Aug-2014.)
𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑𝑈𝐴)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))
 
Theoremlsatcmp2 36022 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 36021. TODO: can lspsncmp 19819 shorten this? (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑 → (𝑈𝐴𝑈 = { 0 }))       (𝜑 → (𝑇𝑈𝑇 = 𝑈))
 
Theoremlsatel 36023 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 36024 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 36025 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 36026* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 36823 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 36027* Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 19831. (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 36028 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29357 analog.) Explicit atom version of lsmcv 19844. (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 36029* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 30063 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈 ≠ { 0 })       (𝜑 → ∃𝑞𝐴 𝑞𝑈)
 
Theoremlssats 36030* 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 30066 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 = (𝑁 {𝑥𝐴𝑥𝑈}))
 
Theoremlpssat 36031* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 30068 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝑈)       (𝜑 → ∃𝑞𝐴 (𝑞𝑈 ∧ ¬ 𝑞𝑇))
 
Theoremlrelat 36032* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 30069 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 36033* The ordering of two subspaces is determined by the atoms under them. (chrelat3 30076 analog.) (Contributed by NM, 29-Oct-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝑈 ↔ ∀𝑝𝐴 (𝑝𝑇𝑝𝑈)))
 
Theoremlssat 36034* 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 30068 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆𝑉𝑆) ∧ 𝑈𝑉) → ∃𝑝𝐴 (𝑝𝑉 ∧ ¬ 𝑝𝑈))
 
Theoremislshpat 36035* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 35998. (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 36036 Extend class notation with the covering relation for a left module or left vector space.
class L
 
Definitiondf-lcv 36037* 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 36039 for binary relation. (df-cv 29984 analog.) (Contributed by NM, 7-Jan-2015.)
L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
 
Theoremlcvfbr 36038* 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 36039* The covers relation for a left vector space (or a left module). (cvbr 29987 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
 
Theoremlcvbr2 36040* The covers relation for a left vector space (or a left module). (cvbr2 29988 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
 
Theoremlcvbr3 36041* 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 36042 The covers relation implies proper subset. (cvpss 29990 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶𝑈)       (𝜑𝑇𝑈)
 
Theoremlcvnbtwn 36043 The covers relation implies no in-betweenness. (cvnbtwn 29991 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)       (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
 
Theoremlcvntr 36044 The covers relation is not transitive. (cvntr 29997 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑇𝐶𝑈)       (𝜑 → ¬ 𝑅𝐶𝑈)
 
Theoremlcvnbtwn2 36045 The covers relation implies no in-betweenness. (cvnbtwn2 29992 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑇)
 
Theoremlcvnbtwn3 36046 The covers relation implies no in-betweenness. (cvnbtwn3 29993 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑅)
 
Theoremlsmcv2 36047 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 29998 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 36048* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 30071 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 36049 An atom covers the zero subspace. (atcv0 30047 analog.) (Contributed by NM, 7-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)       (𝜑 → { 0 }𝐶𝑄)
 
Theoremlsatcveq0 36050 A subspace covered by an atom must be the zero subspace. (atcveq0 30053 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 36051 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 36052 Lemma for lcvexch 36057. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇 ⊊ (𝑇 𝑈) ↔ (𝑇𝑈) ⊊ 𝑈))
 
Theoremlcvexchlem2 36053 Lemma for lcvexch 36057. (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 36054 Lemma for lcvexch 36057. (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 36055 Lemma for lcvexch 36057. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶(𝑇 𝑈))       (𝜑 → (𝑇𝑈)𝐶𝑈)
 
Theoremlcvexchlem5 36056 Lemma for lcvexch 36057. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑 → (𝑇𝑈)𝐶𝑈)       (𝜑𝑇𝐶(𝑇 𝑈))
 
Theoremlcvexch 36057 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 30074 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 36058 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30080 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 36059 Covering property of a subspace plus an atom. (chcv1 30060 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 36060 Covering property of a subspace plus an atom. (chcv2 30061 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 36061 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 30086 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 36062 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 30081 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (¬ 𝑄𝑈 ↔ (𝑈𝑄) = { 0 }))
 
Theoremlsatnem0 36063 The meet of distinct atoms is the zero subspace. (atnemeq0 30082 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)       (𝜑 → (𝑄𝑅 ↔ (𝑄𝑅) = { 0 }))
 
Theoremlsatexch1 36064 The atom exch1ange property. (hlatexch1 36413 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 36065 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 30084 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 36066 Two atoms covering the zero subspace are equal. (atcv1 30085 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 36067 Lemma for lsatcvat 36068. (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 36068 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 30091 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 36069 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 30092 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 36070 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30101 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 36071 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 36072 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 36493 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 36073 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 36494 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 36074 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 37061 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 36072 and l1cvat 36073 to 𝑈𝐻, which in turn change 𝑈𝐻 in islshpcv 36071 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)
 
20.24.7  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 36075 Extend class notation with all linear functionals of a left module or left vector space.
class LFnl
 
Definitiondf-lfl 36076* 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 36077* 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 36078* 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 36079 Property of a linear functional. (lnfnli 29745 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 36080* Properties that determine a linear functional. TODO: use this in place of islfl 36078 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 36081 A linear functional is a function from vectors to scalars. (lnfnfi 29746 analog.) (Contributed by NM, 15-Apr-2014.)
𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (LFnl‘𝑊)       ((𝑊𝑋𝐺𝐹) → 𝐺:𝑉𝐾)
 
Theoremlflcl 36082 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 36083 A linear functional is zero at the zero vector. (lnfn0i 29747 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 36084 Property of a linear functional. (lnfnaddi 29748 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 36085 Property of a linear functional. (lnfnaddi 29748 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 36086 Property of a linear functional. (lnfnmuli 29749 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 36087 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 36088* 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 36089 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 36090 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 36091 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 36092 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 36093* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 36164, 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 36094* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 36164, 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 36095 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 36096 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 36097 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 36098 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 36099 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 36100 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 }))
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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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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