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Theorem List for Metamath Proof Explorer - 35101-35200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremax12indalem 35101 Lemma for ax12inda2 35103 and ax12inda 35104. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))

Theoremax12inda2ALT 35102* Alternate proof of ax12inda2 35103, slightly more direct and not requiring ax-c16 35048. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda2 35103* Induction step for constructing a substitution instance of ax-c15 35045 without using ax-c15 35045. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 35104. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12inda 35104* Induction step for constructing a substitution instance of ax-c15 35045 without using ax-c15 35045. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 35103 may be used instead to avoid the dummy variable 𝑤 in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑤 → (𝑥 = 𝑤 → (𝜑 → ∀𝑥(𝑥 = 𝑤𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))

Theoremax12v2-o 35105* Rederivation of ax-c15 35045 from ax12v 2164 (without using ax-c15 35045 or the full ax-12 2163). Thus, the hypothesis (ax12v 2164) provides an alternate axiom that can be used in place of ax-c15 35045. See also axc15 2387. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12a2-o 35106* Derive ax-c15 35045 from a hypothesis in the form of ax-12 2163, without using ax-12 2163 or ax-c15 35045. The hypothesis is weaker than ax-12 2163, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2163, if we also have ax-c11 35043, which this proof uses. As theorem ax12 2389 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 35044 instead of ax-c11 35043. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremaxc11-o 35107 Show that ax-c11 35043 can be derived from ax-c11n 35044 and ax-12 2163. An open problem is whether this theorem can be derived from ax-c11n 35044 and the others when ax-12 2163 is replaced with ax-c15 35045 or ax12v 2164. See theorem axc11nfromc11 35082 for the rederivation of ax-c11n 35044 from axc11 2396.

Normally, axc11 2396 should be used rather than ax-c11 35043 or axc11-o 35107, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremfsumshftd 35108* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 14916. The proof demonstrates how this can be derived starting from from fsumshft 14916. (Contributed by NM, 1-Nov-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗 = (𝑘𝐾)) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

Axiomax-riotaBAD 35109 Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse 𝐴. See also comments for df-iota 6099. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 6883, CONFLICTS WITH (THE NEW) df-riota 6883 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
(𝑥𝐴 𝜑) = if(∃!𝑥𝐴 𝜑, (℩𝑥(𝑥𝐴𝜑)), (Undef‘{𝑥𝑥𝐴}))

TheoremriotaclbgBAD 35110* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))

TheoremriotaclbBAD 35111* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
𝐴 ∈ V       (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)

Theoremriotasvd 35112* Deduction version of riotasv 35115. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   (𝜑𝐷𝐴)       ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))

Theoremriotasv2d 35113* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5115). Special case of riota2f 6904. (Contributed by NM, 2-Mar-2013.)
𝑦𝜑    &   (𝜑𝑦𝐹)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝑦 = 𝐸) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)    &   (𝜑𝐷𝐴)    &   (𝜑𝐸𝐵)    &   (𝜑𝜒)       ((𝜑𝐴𝑉) → 𝐷 = 𝐹)

Theoremriotasv2s 35114* The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5115) in the form of a substitution instance. Special case of riota2f 6904. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)

Theoremriotasv 35115* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5115). Special case of riota2f 6904. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐴 ∈ V    &   𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)

Theoremriotasv3d 35116* A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5115) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜃)    &   (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   ((𝜑𝐶 = 𝐷) → (𝜒𝜃))    &   (𝜑 → ((𝑦𝐵𝜓) → 𝜒))    &   (𝜑𝐷𝐴)    &   (𝜑 → ∃𝑦𝐵 𝜓)       ((𝜑𝐴𝑉) → 𝜃)

20.23.4  Experiments with weak deduction theorem

Theoremelimhyps 35117 A version of elimhyp 4370 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Theoremdedths 35118 A version of weak deduction theorem dedth 4363 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓       (𝜑𝜓)

TheoremrenegclALT 35119 Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 10686. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremelimhyps2 35120 Generalization of elimhyps 35117 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Theoremdedths2 35121 Generalization of dedths 35118 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theoremnfcxfrdf 35122 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴)

Theoremnfded 35123 A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (𝑥𝐴 {𝑦 ∣ ∀𝑥𝑦𝐴} = 𝐴)) that starts from abidnf 3585. The last is assigned to the inference form (e.g., 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2941. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝑥𝐴𝐵 = 𝐶)    &   𝑥𝐵       (𝜑𝑥𝐶)

Theoremnfded2 35124 A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g., ((𝑥𝐴𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4653) that starts from abidnf 3585. The last is assigned to the inference form (e.g., 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4652) whose hypotheses are satisfied using nfaba1 2941. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)    &   ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)    &   𝑥𝐶       (𝜑𝑥𝐷)

TheoremnfunidALT2 35125 Deduction version of nfuni 4677. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfunidALT 35126 Deduction version of nfuni 4677. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

TheoremnfopdALT 35127 Deduction version of bound-variable hypothesis builder nfop 4652. This shows how the deduction version of a not-free theorem such as nfop 4652 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)

20.23.5  Miscellanea

Theoremcnaddcom 35128 Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremtoycom 35129* Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on . This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of 𝐶. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
𝐶 = {𝑔 ∈ Abel ∣ (Base‘𝑔) = ℂ}    &    + = (+g𝐾)       ((𝐾𝐶𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

20.23.6  Atoms, hyperplanes, and covering in a left vector space (or module)

Syntaxclsa 35130 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms

Syntaxclsh 35131 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp

Definitiondf-lsatoms 35132* Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))

Definitiondf-lshyp 35133* Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.)
LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})

Theoremlshpset 35134* The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)       (𝑊𝑋𝐻 = {𝑠𝑆 ∣ (𝑠𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Theoremislshp 35135* The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)       (𝑊𝑋 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Theoremislshpsm 35136* Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 → (𝑈𝐻 ↔ (𝑈𝑆𝑈𝑉 ∧ ∃𝑣𝑉 (𝑈 (𝑁‘{𝑣})) = 𝑉)))

Theoremlshplss 35137 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑆)

Theoremlshpne 35138 A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑉)

Theoremlshpnel 35139 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 35140 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 35141 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 35142 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 35143 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 35144 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 35145 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 35146* 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 35147* 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 35148 The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 35149 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 35149 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 35150* 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 35151* 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 35152 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 35153 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈𝑆)

Theoremlsatssv 35154 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)       (𝜑𝑄𝑉)

Theoremlsatn0 35155 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 29776 analog.) (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈 ≠ { 0 })

Theoremlsatspn0 35156 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 35157 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 35158 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 35159 If two atoms are comparable, they are equal. (atsseq 29778 analog.) TODO: can lspsncmp 19511 shorten this? (Contributed by NM, 25-Aug-2014.)
𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑𝑈𝐴)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

Theoremlsatcmp2 35160 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 35159. TODO: can lspsncmp 19511 shorten this? (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑 → (𝑈𝐴𝑈 = { 0 }))       (𝜑 → (𝑇𝑈𝑇 = 𝑈))

Theoremlsatel 35161 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 35162 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 35163 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 35164* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 35961 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 35165* Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 19523. (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 35166 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29083 analog.) Explicit atom version of lsmcv 19537. (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 35167* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 29789 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈 ≠ { 0 })       (𝜑 → ∃𝑞𝐴 𝑞𝑈)

Theoremlssats 35168* 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 29792 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 = (𝑁 {𝑥𝐴𝑥𝑈}))

Theoremlpssat 35169* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 29794 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝑈)       (𝜑 → ∃𝑞𝐴 (𝑞𝑈 ∧ ¬ 𝑞𝑇))

Theoremlrelat 35170* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 29795 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 35171* The ordering of two subspaces is determined by the atoms under them. (chrelat3 29802 analog.) (Contributed by NM, 29-Oct-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝑈 ↔ ∀𝑝𝐴 (𝑝𝑇𝑝𝑈)))

Theoremlssat 35172* 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 29794 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆𝑉𝑆) ∧ 𝑈𝑉) → ∃𝑝𝐴 (𝑝𝑉 ∧ ¬ 𝑝𝑈))

Theoremislshpat 35173* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 35136. (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 35174 Extend class notation with the covering relation for a left module or left vector space.
class L

Definitiondf-lcv 35175* 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 35177 for binary relation. (df-cv 29710 analog.) (Contributed by NM, 7-Jan-2015.)
L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})

Theoremlcvfbr 35176* 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 35177* The covers relation for a left vector space (or a left module). (cvbr 29713 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))

Theoremlcvbr2 35178* The covers relation for a left vector space (or a left module). (cvbr2 29714 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))

Theoremlcvbr3 35179* 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 35180 The covers relation implies proper subset. (cvpss 29716 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶𝑈)       (𝜑𝑇𝑈)

Theoremlcvnbtwn 35181 The covers relation implies no in-betweenness. (cvnbtwn 29717 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)       (𝜑 → ¬ (𝑅𝑈𝑈𝑇))

Theoremlcvntr 35182 The covers relation is not transitive. (cvntr 29723 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑇𝐶𝑈)       (𝜑 → ¬ 𝑅𝐶𝑈)

Theoremlcvnbtwn2 35183 The covers relation implies no in-betweenness. (cvnbtwn2 29718 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑇)

Theoremlcvnbtwn3 35184 The covers relation implies no in-betweenness. (cvnbtwn3 29719 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑅)

Theoremlsmcv2 35185 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 29724 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 35186* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 29797 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 35187 An atom covers the zero subspace. (atcv0 29773 analog.) (Contributed by NM, 7-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)       (𝜑 → { 0 }𝐶𝑄)

Theoremlsatcveq0 35188 A subspace covered by an atom must be the zero subspace. (atcveq0 29779 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 35189 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 35190 Lemma for lcvexch 35195. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇 ⊊ (𝑇 𝑈) ↔ (𝑇𝑈) ⊊ 𝑈))

Theoremlcvexchlem2 35191 Lemma for lcvexch 35195. (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 35192 Lemma for lcvexch 35195. (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 35193 Lemma for lcvexch 35195. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶(𝑇 𝑈))       (𝜑 → (𝑇𝑈)𝐶𝑈)

Theoremlcvexchlem5 35194 Lemma for lcvexch 35195. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑 → (𝑇𝑈)𝐶𝑈)       (𝜑𝑇𝐶(𝑇 𝑈))

Theoremlcvexch 35195 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 29800 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 35196 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 29806 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 35197 Covering property of a subspace plus an atom. (chcv1 29786 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 35198 Covering property of a subspace plus an atom. (chcv2 29787 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 35199 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 29812 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 35200 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 29807 analog.) (Contributed by NM, 10-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑄𝐴)       (𝜑 → (¬ 𝑄𝑈 ↔ (𝑈𝑄) = { 0 }))

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