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Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax12eq 36601 Basis step for constructing a substitution instance of ax-c15 36549 without using ax-c15 36549. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦𝑧 = 𝑤))))
 
Theoremax12el 36602 Basis step for constructing a substitution instance of ax-c15 36549 without using ax-c15 36549. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧𝑤 → ∀𝑥(𝑥 = 𝑦𝑧𝑤))))
 
Theoremax12indn 36603 Induction step for constructing a substitution instance of ax-c15 36549 without using ax-c15 36549. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))
 
Theoremax12indi 36604 Induction step for constructing a substitution instance of ax-c15 36549 without using ax-c15 36549. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜓))))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))))
 
Theoremax12indalem 36605 Lemma for ax12inda2 36607 and ax12inda 36608. (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 36606* Alternate proof of ax12inda2 36607, slightly more direct and not requiring ax-c16 36552. (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 36607* Induction step for constructing a substitution instance of ax-c15 36549 without using ax-c15 36549. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 36608. (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 36608* Induction step for constructing a substitution instance of ax-c15 36549 without using ax-c15 36549. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 36607 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 36609* Rederivation of ax-c15 36549 from ax12v 2180 (without using ax-c15 36549 or the full ax-12 2179). Thus, the hypothesis (ax12v 2180) provides an alternate axiom that can be used in place of ax-c15 36549. See also axc15 2423. (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 36610* Derive ax-c15 36549 from a hypothesis in the form of ax-12 2179, without using ax-12 2179 or ax-c15 36549. The hypothesis is weaker than ax-12 2179, 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 2179, if we also have ax-c11 36547, which this proof uses. As Theorem ax12 2424 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 36548 instead of ax-c11 36547. (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 36611 Show that ax-c11 36547 can be derived from ax-c11n 36548 and ax-12 2179. An open problem is whether this theorem can be derived from ax-c11n 36548 and the others when ax-12 2179 is replaced with ax-c15 36549 or ax12v 2180. See Theorems axc11nfromc11 36586 for the rederivation of ax-c11n 36548 from axc11 2431.

Normally, axc11 2431 should be used rather than ax-c11 36547 or axc11-o 36611, 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 36612* Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 15231. The proof demonstrates how this can be derived starting from from fsumshft 15231. (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 36613 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 6298. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 7130, CONFLICTS WITH (THE NEW) df-riota 7130 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 36614* Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝐴𝑉 → (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴))
 
TheoremriotaclbBAD 36615* Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
𝐴 ∈ V       (∃!𝑥𝐴 𝜑 ↔ (𝑥𝐴 𝜑) ∈ 𝐴)
 
Theoremriotasvd 36616* Deduction version of riotasv 36619. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))    &   (𝜑𝐷𝐴)       ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
 
Theoremriotasv2d 36617* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5271). Special case of riota2f 7155. (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 36618* The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5271) in the form of a substitution instance. Special case of riota2f 7155. (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 36619* Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5271). Special case of riota2f 7155. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
𝐴 ∈ V    &   𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))       ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
 
Theoremriotasv3d 36620* A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5271) 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)
 
20.24.4  Experiments with weak deduction theorem
 
Theoremelimhyps 36621 A version of elimhyp 4480 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[𝐵 / 𝑥]𝜑       [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
 
Theoremdedths 36622 A version of weak deduction theorem dedth 4473 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓       (𝜑𝜓)
 
TheoremrenegclALT 36623 Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 11030. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
 
Theoremelimhyps2 36624 Generalization of elimhyps 36621 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 36625 Generalization of dedths 36622 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 36626 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 36627 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 3603. The last is assigned to the inference form (e.g., 𝑥 {𝑦 ∣ ∀𝑥𝑦𝐴}) whose hypothesis is satisfied using nfaba1 2908. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝑥𝐴𝐵 = 𝐶)    &   𝑥𝐵       (𝜑𝑥𝐶)
 
Theoremnfded2 36628 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 4779) that starts from abidnf 3603. The last is assigned to the inference form (e.g., 𝑥⟨{𝑦 ∣ ∀𝑥𝑦𝐴}, {𝑦 ∣ ∀𝑥𝑦𝐵}⟩ for nfop 4778) whose hypotheses are satisfied using nfaba1 2908. (Contributed by NM, 19-Nov-2020.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)    &   ((𝑥𝐴𝑥𝐵) → 𝐶 = 𝐷)    &   𝑥𝐶       (𝜑𝑥𝐷)
 
TheoremnfunidALT2 36629 Deduction version of nfuni 4804. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
TheoremnfunidALT 36630 Deduction version of nfuni 4804. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
TheoremnfopdALT 36631 Deduction version of bound-variable hypothesis builder nfop 4778. This shows how the deduction version of a not-free theorem such as nfop 4778 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⟩)
 
20.24.5  Miscellanea
 
Theoremcnaddcom 36632 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 36633* 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))
 
20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 36634 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms
 
Syntaxclsh 36635 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp
 
Definitiondf-lsatoms 36636* 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 36637* 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 than 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 36638* 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 36639* 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 36640* 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 36641 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐻)       (𝜑𝑈𝑆)
 
Theoremlshpne 36642 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 36643 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 36644 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 36645 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 36646 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 36647 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 36648 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 36649 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 36650* 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 36651* 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 36652 The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 36653 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 36653 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 36654* 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 36655* 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 36656 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 36657 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈𝑆)
 
Theoremlsatssv 36658 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑄𝐴)       (𝜑𝑄𝑉)
 
Theoremlsatn0 36659 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 30283 analog.) (Contributed by NM, 25-Aug-2014.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝐴)       (𝜑𝑈 ≠ { 0 })
 
Theoremlsatspn0 36660 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 36661 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 36662 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 36663 If two atoms are comparable, they are equal. (atsseq 30285 analog.) TODO: can lspsncmp 20010 shorten this? (Contributed by NM, 25-Aug-2014.)
𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑𝑈𝐴)       (𝜑 → (𝑇𝑈𝑇 = 𝑈))
 
Theoremlsatcmp2 36664 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 36663. TODO: can lspsncmp 20010 shorten this? (Contributed by NM, 3-Feb-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝐴)    &   (𝜑 → (𝑈𝐴𝑈 = { 0 }))       (𝜑 → (𝑇𝑈𝑇 = 𝑈))
 
Theoremlsatel 36665 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 36666 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 36667 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 36668* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 37465 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 36669* Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 20022. (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 36670 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29590 analog.) Explicit atom version of lsmcv 20035. (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 36671* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 30296 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑈 ≠ { 0 })       (𝜑 → ∃𝑞𝐴 𝑞𝑈)
 
Theoremlssats 36672* 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 30299 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 = (𝑁 {𝑥𝐴𝑥𝑈}))
 
Theoremlpssat 36673* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 30301 analog.) (Contributed by NM, 11-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝑈)       (𝜑 → ∃𝑞𝐴 (𝑞𝑈 ∧ ¬ 𝑞𝑇))
 
Theoremlrelat 36674* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 30302 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 36675* The ordering of two subspaces is determined by the atoms under them. (chrelat3 30309 analog.) (Contributed by NM, 29-Oct-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝑈 ↔ ∀𝑝𝐴 (𝑝𝑇𝑝𝑈)))
 
Theoremlssat 36676* 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 30301 analog.) (Contributed by NM, 9-Apr-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝐴 = (LSAtoms‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆𝑉𝑆) ∧ 𝑈𝑉) → ∃𝑝𝐴 (𝑝𝑉 ∧ ¬ 𝑝𝑈))
 
Theoremislshpat 36677* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 36640. (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 36678 Extend class notation with the covering relation for a left module or left vector space.
class L
 
Definitiondf-lcv 36679* 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 36681 for binary relation. (df-cv 30217 analog.) (Contributed by NM, 7-Jan-2015.)
L = (𝑤 ∈ V ↦ {⟨𝑡, 𝑢⟩ ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡𝑠𝑠𝑢)))})
 
Theoremlcvfbr 36680* 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 36681* The covers relation for a left vector space (or a left module). (cvbr 30220 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ¬ ∃𝑠𝑆 (𝑇𝑠𝑠𝑈))))
 
Theoremlcvbr2 36682* The covers relation for a left vector space (or a left module). (cvbr2 30221 analog.) (Contributed by NM, 9-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇𝑈 ∧ ∀𝑠𝑆 ((𝑇𝑠𝑠𝑈) → 𝑠 = 𝑈))))
 
Theoremlcvbr3 36683* 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 36684 The covers relation implies proper subset. (cvpss 30223 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶𝑈)       (𝜑𝑇𝑈)
 
Theoremlcvnbtwn 36685 The covers relation implies no in-betweenness. (cvnbtwn 30224 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)       (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
 
Theoremlcvntr 36686 The covers relation is not transitive. (cvntr 30230 analog.) (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑇𝐶𝑈)       (𝜑 → ¬ 𝑅𝐶𝑈)
 
Theoremlcvnbtwn2 36687 The covers relation implies no in-betweenness. (cvnbtwn2 30225 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑇)
 
Theoremlcvnbtwn3 36688 The covers relation implies no in-betweenness. (cvnbtwn3 30226 analog.) (Contributed by NM, 7-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑅𝑆)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑅𝐶𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑈𝑇)       (𝜑𝑈 = 𝑅)
 
Theoremlsmcv2 36689 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 30231 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 36690* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 30304 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 36691 An atom covers the zero subspace. (atcv0 30280 analog.) (Contributed by NM, 7-Jan-2015.)
0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑄𝐴)       (𝜑 → { 0 }𝐶𝑄)
 
Theoremlsatcveq0 36692 A subspace covered by an atom must be the zero subspace. (atcveq0 30286 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 36693 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 36694 Lemma for lcvexch 36699. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑇 ⊊ (𝑇 𝑈) ↔ (𝑇𝑈) ⊊ 𝑈))
 
Theoremlcvexchlem2 36695 Lemma for lcvexch 36699. (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 36696 Lemma for lcvexch 36699. (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 36697 Lemma for lcvexch 36699. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑇𝐶(𝑇 𝑈))       (𝜑 → (𝑇𝑈)𝐶𝑈)
 
Theoremlcvexchlem5 36698 Lemma for lcvexch 36699. (Contributed by NM, 10-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐶 = ( ⋖L𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑 → (𝑇𝑈)𝐶𝑈)       (𝜑𝑇𝐶(𝑇 𝑈))
 
Theoremlcvexch 36699 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 30307 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 36700 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30313 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)))
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