P. 385 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38401-38500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lsatlspsn2 38401	The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 38402 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋}) ∈ 𝐴)
Theorem	lsatlspsn 38402	The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
Theorem	islsati 38403*	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)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑈 ∈ 𝐴) → ∃𝑣 ∈ 𝑉 𝑈 = (𝑁‘{𝑣}))
Theorem	lsateln0 38404*	A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 )
Theorem	lsatlss 38405	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 → 𝐴 ⊆ 𝑆)
Theorem	lsatlssel 38406	An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝑆)
Theorem	lsatssv 38407	An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑄 ⊆ 𝑉)
Theorem	lsatn0 38408	A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 32142 analog.) (Contributed by NM, 25-Aug-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑈 ≠ { 0 })
Theorem	lsatspn0 38409	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 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ 𝑋 ≠ 0 ))
Theorem	lsator0sp 38410	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 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ∨ (𝑁‘{𝑋}) = { 0 }))
Theorem	lsatssn0 38411	A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝑈 ≠ { 0 })
Theorem	lsatcmp 38412	If two atoms are comparable, they are equal. (atsseq 32144 analog.) TODO: can lspsncmp 20993 shorten this? (Contributed by NM, 25-Aug-2014.)
⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈))
Theorem	lsatcmp2 38413	If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 38412. TODO: can lspsncmp 20993 shorten this? (Contributed by NM, 3-Feb-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∨ 𝑈 = { 0 }))    ⇒   ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈))
Theorem	lsatel 38414	A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → 𝑈 = (𝑁‘{𝑋}))
Theorem	lsatelbN 38415	A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑈 = (𝑁‘{𝑋})))
Theorem	lsat2el 38416	Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑋 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑄)    ⇒   ⊢ (𝜑 → 𝑃 = 𝑄)
Theorem	lsmsat 38417*	Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 39215 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ≠ { 0 })    &   ⊢ (𝜑 → 𝑄 ⊆ (𝑇 ⊕ 𝑈))    ⇒   ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))
Theorem	lsatfixedN 38418*	Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 21005. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑋}))    &   ⊢ (𝜑 → 𝑄 ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → 𝑄 ⊆ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑌}) ∖ { 0 })𝑄 = (𝑁‘{(𝑋 + 𝑧)}))
Theorem	lsmsatcv 38419	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31449 analog.) Explicit atom version of lsmcv 21018. (Contributed by NM, 29-Oct-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄))
Theorem	lssatomic 38420*	The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 32155 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ≠ { 0 })    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 𝑞 ⊆ 𝑈)
Theorem	lssats 38421*	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 32158 analog.) (Contributed by NM, 9-Apr-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 = (𝑁‘∪ {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝑈}))
Theorem	lpssat 38422*	Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 32160 analog.) (Contributed by NM, 11-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ⊊ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇))
Theorem	lrelat 38423*	Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 32161 analog.) (Contributed by NM, 11-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ⊊ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))
Theorem	lssatle 38424*	The ordering of two subspaces is determined by the atoms under them. (chrelat3 32168 analog.) (Contributed by NM, 29-Oct-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈)))
Theorem	lssat 38425*	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 32160 analog.) (Contributed by NM, 9-Apr-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑉 ∈ 𝑆) ∧ 𝑈 ⊊ 𝑉) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ 𝑈))
Theorem	islshpat 38426*	Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 38389. (Contributed by NM, 11-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑞 ∈ 𝐴 (𝑈 ⊕ 𝑞) = 𝑉)))
Syntax	clcv 38427	Extend class notation with the covering relation for a left module or left vector space.
class ⋖L
Definition	df-lcv 38428*	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( ⋖L ‘𝑊)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 38430 for binary relation. (df-cv 32076 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ ⋖L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
Theorem	lcvfbr 38429*	The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐶 = {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))})
Theorem	lcvbr 38430*	The covers relation for a left vector space (or a left module). (cvbr 32079 analog.) (Contributed by NM, 9-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))))
Theorem	lcvbr2 38431*	The covers relation for a left vector space (or a left module). (cvbr2 32080 analog.) (Contributed by NM, 9-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈) → 𝑠 = 𝑈))))
Theorem	lcvbr3 38432*	The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = 𝑇 ∨ 𝑠 = 𝑈)))))
Theorem	lcvpss 38433	The covers relation implies proper subset. (cvpss 32082 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇𝐶𝑈)    ⇒   ⊢ (𝜑 → 𝑇 ⊊ 𝑈)
Theorem	lcvnbtwn 38434	The covers relation implies no in-betweenness. (cvnbtwn 32083 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅𝐶𝑇)    ⇒   ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
Theorem	lcvntr 38435	The covers relation is not transitive. (cvntr 32089 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅𝐶𝑇)    &   ⊢ (𝜑 → 𝑇𝐶𝑈)    ⇒   ⊢ (𝜑 → ¬ 𝑅𝐶𝑈)
Theorem	lcvnbtwn2 38436	The covers relation implies no in-betweenness. (cvnbtwn2 32084 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅𝐶𝑇)    &   ⊢ (𝜑 → 𝑅 ⊊ 𝑈)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑇)    ⇒   ⊢ (𝜑 → 𝑈 = 𝑇)
Theorem	lcvnbtwn3 38437	The covers relation implies no in-betweenness. (cvnbtwn3 32085 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅𝐶𝑇)    &   ⊢ (𝜑 → 𝑅 ⊆ 𝑈)    &   ⊢ (𝜑 → 𝑈 ⊊ 𝑇)    ⇒   ⊢ (𝜑 → 𝑈 = 𝑅)
Theorem	lsmcv2 38438	Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 32090 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝑈𝐶(𝑈 ⊕ (𝑁‘{𝑋})))
Theorem	lcvat 38439*	If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 32163 analog.) (Contributed by NM, 11-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇𝐶𝑈)    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊕ 𝑞) = 𝑈)
Theorem	lsatcv0 38440	An atom covers the zero subspace. (atcv0 32139 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → { 0 }𝐶𝑄)
Theorem	lsatcveq0 38441	A subspace covered by an atom must be the zero subspace. (atcveq0 32145 analog.) (Contributed by NM, 7-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑈𝐶𝑄 ↔ 𝑈 = { 0 }))
Theorem	lsat0cv 38442	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 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ { 0 }𝐶𝑈))
Theorem	lcvexchlem1 38443	Lemma for lcvexch 38448. (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
Theorem	lcvexchlem2 38444	Lemma for lcvexch 38448. (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑅)    &   ⊢ (𝜑 → 𝑅 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑅 ⊕ 𝑇) ∩ 𝑈) = 𝑅)
Theorem	lcvexchlem3 38445	Lemma for lcvexch 38448. (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑅)    &   ⊢ (𝜑 → 𝑅 ⊆ (𝑇 ⊕ 𝑈))    ⇒   ⊢ (𝜑 → ((𝑅 ∩ 𝑈) ⊕ 𝑇) = 𝑅)
Theorem	lcvexchlem4 38446	Lemma for lcvexch 38448. (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))    ⇒   ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)
Theorem	lcvexchlem5 38447	Lemma for lcvexch 38448. (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)    ⇒   ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))
Theorem	lcvexch 38448	Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 32166 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ 𝑇𝐶(𝑇 ⊕ 𝑈)))
Theorem	lcvp 38449	Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 32172 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑈 ∩ 𝑄) = { 0 } ↔ 𝑈𝐶(𝑈 ⊕ 𝑄)))
Theorem	lcv1 38450	Covering property of a subspace plus an atom. (chcv1 32152 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈𝐶(𝑈 ⊕ 𝑄)))
Theorem	lcv2 38451	Covering property of a subspace plus an atom. (chcv2 32153 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑈 ⊊ (𝑈 ⊕ 𝑄) ↔ 𝑈𝐶(𝑈 ⊕ 𝑄)))
Theorem	lsatexch 38452	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 32178 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))    &   ⊢ (𝜑 → (𝑈 ∩ 𝑄) = { 0 })    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄))
Theorem	lsatnle 38453	The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 32173 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ (𝑈 ∩ 𝑄) = { 0 }))
Theorem	lsatnem0 38454	The meet of distinct atoms is the zero subspace. (atnemeq0 32174 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 }))
Theorem	lsatexch1 38455	The atom exch1ange property. (hlatexch1 38805 analog.) (Contributed by NM, 14-Jan-2015.)
⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ⊆ (𝑆 ⊕ 𝑅))    &   ⊢ (𝜑 → 𝑄 ≠ 𝑆)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (𝑆 ⊕ 𝑄))
Theorem	lsatcv0eq 38456	If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 32176 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ({ 0 }𝐶(𝑄 ⊕ 𝑅) ↔ 𝑄 = 𝑅))
Theorem	lsatcv1 38457	Two atoms covering the zero subspace are equal. (atcv1 32177 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅))    ⇒   ⊢ (𝜑 → (𝑈 = { 0 } ↔ 𝑄 = 𝑅))
Theorem	lsatcvatlem 38458	Lemma for lsatcvat 38459. (Contributed by NM, 10-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈 ≠ { 0 })    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))    &   ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝐴)
Theorem	lsatcvat 38459	A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32183 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈 ≠ { 0 })    &   ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝐴)
Theorem	lsatcvat2 38460	A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 32184 analog.) (Contributed by NM, 10-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    &   ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅))    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝐴)
Theorem	lsatcvat3 38461	A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32193 analog.) (Contributed by NM, 11-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    &   ⊢ (𝜑 → ¬ 𝑅 ⊆ 𝑈)    &   ⊢ (𝜑 → 𝑄 ⊆ (𝑈 ⊕ 𝑅))    ⇒   ⊢ (𝜑 → (𝑈 ∩ (𝑄 ⊕ 𝑅)) ∈ 𝐴)
Theorem	islshpcv 38462	Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    ⇒   ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈𝐶𝑉)))
Theorem	l1cvpat 38463	A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 38885 analog.) (Contributed by NM, 11-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑈𝐶𝑉)    &   ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = 𝑉)
Theorem	l1cvat 38464	Create an atom under an element covered by the lattice unity. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 38886 analog.) (Contributed by NM, 11-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐶 = ( ⋖L ‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    &   ⊢ (𝜑 → 𝑈𝐶𝑉)    &   ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴)
Theorem	lshpat 38465	Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 39453 analog.) TODO: This changes 𝑈𝐶𝑉 in l1cvpat 38463 and l1cvat 38464 to 𝑈 ∈ 𝐻, which in turn change 𝑈 ∈ 𝐻 in islshpcv 38462 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)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    &   ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴)
21.26.7  Functionals and kernels of a left vector space (or module)
Syntax	clfn 38466	Extend class notation with all linear functionals of a left module or left vector space.
class LFnl
Definition	df-lfl 38467*	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( ·𝑠 ‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦))})
Theorem	lflset 38468*	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)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ × = (.r‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐹 = {𝑓 ∈ (𝐾 ↑m 𝑉) ∣ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑓‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝑓‘𝑥)) ⨣ (𝑓‘𝑦))})
Theorem	islfl 38469*	The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ × = (.r‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))))
Theorem	lfli 38470	Property of a linear functional. (lnfnli 31837 analog.) (Contributed by NM, 16-Apr-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ × = (.r‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))
Theorem	islfld 38471*	Properties that determine a linear functional. TODO: use this in place of islfl 38469 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
⊢ (𝜑 → 𝑉 = (Base‘𝑊))    &   ⊢ (𝜑 → + = (+g‘𝑊))    &   ⊢ (𝜑 → 𝐷 = (Scalar‘𝑊))    &   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝐷))    &   ⊢ (𝜑 → ⨣ = (+g‘𝐷))    &   ⊢ (𝜑 → × = (.r‘𝐷))    &   ⊢ (𝜑 → 𝐹 = (LFnl‘𝑊))    &   ⊢ (𝜑 → 𝐺:𝑉⟶𝐾)    &   ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝐹)
Theorem	lflf 38472	A linear functional is a function from vectors to scalars. (lnfnfi 31838 analog.) (Contributed by NM, 15-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾)
Theorem	lflcl 38473	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)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾)
Theorem	lfl0 38474	A linear functional is zero at the zero vector. (lnfn0i 31839 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝑍 = (0g‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘𝑍) = 0 )
Theorem	lfladd 38475	Property of a linear functional. (lnfnaddi 31840 analog.) (Contributed by NM, 18-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) ⨣ (𝐺‘𝑌)))
Theorem	lflsub 38476	Property of a linear functional. (lnfnaddi 31840 analog.) (Contributed by NM, 18-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝑀 = (-g‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘(𝑋 − 𝑌)) = ((𝐺‘𝑋)𝑀(𝐺‘𝑌)))
Theorem	lflmul 38477	Property of a linear functional. (lnfnmuli 31841 analog.) (Contributed by NM, 16-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐷)    &   ⊢ × = (.r‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘(𝑅 · 𝑋)) = (𝑅 × (𝐺‘𝑋)))
Theorem	lfl0f 38478	The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ 𝐹)
Theorem	lfl1 38479*	A nonzero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 1 = (1r‘𝐷)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × { 0 })) → ∃𝑥 ∈ 𝑉 (𝐺‘𝑥) = 1 )
Theorem	lfladdcl 38480	Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f + 𝐻) ∈ 𝐹)
Theorem	lfladdcom 38481	Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f + 𝐻) = (𝐻 ∘f + 𝐺))
Theorem	lfladdass 38482	Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f + 𝐼) = (𝐺 ∘f + (𝐻 ∘f + 𝐼)))
Theorem	lfladd0l 38483	Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f + 𝐺) = 𝐺)
Theorem	lflnegcl 38484*	Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 38555, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → 𝑁 ∈ 𝐹)
Theorem	lflnegl 38485*	A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 38555, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → (𝑁 ∘f + 𝐺) = (𝑉 × { 0 }))
Theorem	lflvscl 38486	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)    &   ⊢ · = (.r‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹)
Theorem	lflvsdi1 38487	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝐺 ∘f + 𝐻) ∘f · (𝑉 × {𝑋})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐻 ∘f · (𝑉 × {𝑋}))))
Theorem	lflvsdi2 38488	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f + (𝑉 × {𝑌}))) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))
Theorem	lflvsdi2a 38489	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 + 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f + (𝐺 ∘f · (𝑉 × {𝑌}))))
Theorem	lflvsass 38490	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 · 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f · (𝑉 × {𝑌})))
Theorem	lfl0sc 38491	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)    &   ⊢ · = (.r‘𝐷)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × { 0 })) = (𝑉 × { 0 }))
Theorem	lflsc0N 38492	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)    &   ⊢ · = (.r‘𝐷)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑉 × { 0 }) ∘f · (𝑉 × {𝑋})) = (𝑉 × { 0 }))
Theorem	lfl1sc 38493	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)    &   ⊢ · = (.r‘𝐷)    &   ⊢ 1 = (1r‘𝐷)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × { 1 })) = 𝐺)
Syntax	clk 38494	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
Definition	df-lkr 38495*	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))})))
Theorem	lkrfval 38496*	The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐾 = (𝑓 ∈ 𝐹 ↦ (◡𝑓 “ { 0 })))
Theorem	lkrval 38497	Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = (◡𝐺 “ { 0 }))
Theorem	ellkr 38498	Membership in the kernel of a functional. (elnlfn 31725 analog.) (Contributed by NM, 16-Apr-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 0 )))
Theorem	lkrval2 38499*	Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ 𝑉 ∣ (𝐺‘𝑥) = 0 })
Theorem	ellkr2 38500	Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐹 = (LFnl‘𝑊)    &   ⊢ 𝐾 = (LKer‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝐺) ↔ (𝐺‘𝑋) = 0 ))