P. 384 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38301-38400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ax12eq 38301	Basis step for constructing a substitution instance of ax-c15 38249 without using ax-c15 38249. 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 = 𝑦 → 𝑧 = 𝑤))))
Theorem	ax12el 38302	Basis step for constructing a substitution instance of ax-c15 38249 without using ax-c15 38249. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝑧 ∈ 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑤))))
Theorem	ax12indn 38303	Induction step for constructing a substitution instance of ax-c15 38249 without using ax-c15 38249. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (¬ 𝜑 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))))
Theorem	ax12indi 38304	Induction step for constructing a substitution instance of ax-c15 38249 without using ax-c15 38249. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ((𝜑 → 𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))))
Theorem	ax12indalem 38305	Lemma for ax12inda2 38307 and ax12inda 38308. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
Theorem	ax12inda2ALT 38306*	Alternate proof of ax12inda2 38307, slightly more direct and not requiring ax-c16 38252. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Theorem	ax12inda2 38307*	Induction step for constructing a substitution instance of ax-c15 38249 without using ax-c15 38249. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 38308. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Theorem	ax12inda 38308*	Induction step for constructing a substitution instance of ax-c15 38249 without using ax-c15 38249. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 38307 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 = 𝑤 → 𝜑))))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
Theorem	ax12v2-o 38309*	Rederivation of ax-c15 38249 from ax12v 2164 (without using ax-c15 38249 or the full ax-12 2163). Thus, the hypothesis (ax12v 2164) provides an alternate axiom that can be used in place of ax-c15 38249. See also axc15 2413. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	ax12a2-o 38310*	Derive ax-c15 38249 from a hypothesis in the form of ax-12 2163, without using ax-12 2163 or ax-c15 38249. 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 38247, which this proof uses. As Theorem ax12 2414 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 38248 instead of ax-c11 38247. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	axc11-o 38311	Show that ax-c11 38247 can be derived from ax-c11n 38248 and ax-12 2163. An open problem is whether this theorem can be derived from ax-c11n 38248 and the others when ax-12 2163 is replaced with ax-c15 38249 or ax12v 2164. See Theorems axc11nfromc11 38286 for the rederivation of ax-c11n 38248 from axc11 2421. Normally, axc11 2421 should be used rather than ax-c11 38247 or axc11-o 38311, 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))
Theorem	fsumshftd 38312*	Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 15723. The proof demonstrates how this can be derived starting from from fsumshft 15723. (Contributed by NM, 1-Nov-2019.)
⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑗 = (𝑘 − 𝐾)) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)
Axiom	ax-riotaBAD 38313	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 6485. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 7357, CONFLICTS WITH (THE NEW) df-riota 7357 AND MAKES THE SYSTEM IN set.mm INCONSISTENT. IT IS TEMPORARY AND WILL BE DELETED AFTER ALL USES ARE ELIMINATED.
⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴}))
Theorem	riotaclbgBAD 38314*	Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴))
Theorem	riotaclbBAD 38315*	Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasvd 38316*	Deduction version of riotasv 38319. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
Theorem	riotasv2d 38317*	Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5391). Special case of riota2f 7382. (Contributed by NM, 2-Mar-2013.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝐹)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐸) → 𝐶 = 𝐹)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = 𝐹)
Theorem	riotasv2s 38318*	The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5391) in the form of a substitution instance. Special case of riota2f 7382. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)
Theorem	riotasv 38319*	Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5391). Special case of riota2f 7382. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))    ⇒   ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)
Theorem	riotasv3d 38320*	A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5391) 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.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜃)    &   ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))    &   ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    &   ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)
21.26.4  Experiments with weak deduction theorem
Theorem	elimhyps 38321	A version of elimhyp 4585 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
⊢ [𝐵 / 𝑥]𝜑    ⇒   ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
Theorem	dedths 38322	A version of weak deduction theorem dedth 4578 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	renegclALT 38323	Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 11520. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
Theorem	elimhyps2 38324	Generalization of elimhyps 38321 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
Theorem	dedths2 38325	Generalization of dedths 38322 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)
Theorem	nfcxfrdf 38326	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfded 38327	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., (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3690. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2903. (Contributed by NM, 19-Nov-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶)    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐶)
Theorem	nfded2 38328	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., ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ = ⟨𝐴, 𝐵⟩) for nfopd 4882) that starts from abidnf 3690. The last is assigned to the inference form (e.g., Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4881) whose hypotheses are satisfied using nfaba1 2903. (Contributed by NM, 19-Nov-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷)    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐷)
Theorem	nfunidALT2 38329	Deduction version of nfuni 4906. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	nfunidALT 38330	Deduction version of nfuni 4906. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	nfopdALT 38331	Deduction version of bound-variable hypothesis builder nfop 4881. This shows how the deduction version of a not-free theorem such as nfop 4881 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
21.26.5  Miscellanea
Theorem	cnaddcom 38332	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))
Theorem	toycom 38333*	Show the commutative law for an operation 𝑂 on a toy structure class 𝐶 of commuatitive operations on ℂ. 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}    &   ⊢ + = (+g‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐶 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
21.26.6  Atoms, hyperplanes, and covering in a left vector space (or module)
Syntax	clsa 38334	Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms
Syntax	clsh 38335	Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp
Definition	df-lsatoms 38336*	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) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
Definition	df-lshyp 38337*	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))})
Theorem	lshpset 38338*	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)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
Theorem	islshp 38339*	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)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
Theorem	islshpsm 38340*	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑈 ⊕ (𝑁‘{𝑣})) = 𝑉)))
Theorem	lshplss 38341	A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝑆)
Theorem	lshpne 38342	A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    ⇒   ⊢ (𝜑 → 𝑈 ≠ 𝑉)
Theorem	lshpnel 38343	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
Theorem	lshpnelb 38344	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)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉))
Theorem	lshpnel2N 38345	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)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉))
Theorem	lshpne0 38346	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 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)    ⇒   ⊢ (𝜑 → 𝑋 ≠ 0 )
Theorem	lshpdisj 38347	A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) = 𝑉)    ⇒   ⊢ (𝜑 → (𝑈 ∩ (𝑁‘{𝑋})) = { 0 })
Theorem	lshpcmp 38348	If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    ⇒   ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ 𝑇 = 𝑈))
Theorem	lshpinN 38349	The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐻)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐻)    ⇒   ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ∈ 𝐻 ↔ 𝑇 = 𝑈))
Theorem	lsatset 38350*	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 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
Theorem	islsat 38351*	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 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑈 = (𝑁‘{𝑥})))
Theorem	lsatlspsn2 38352	The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 38353 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 38353	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 38354*	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 38355*	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 38356	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 38357	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 38358	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 38359	A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 32067 analog.) (Contributed by NM, 25-Aug-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑈 ≠ { 0 })
Theorem	lsatspn0 38360	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 38361	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 38362	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 38363	If two atoms are comparable, they are equal. (atsseq 32069 analog.) TODO: can lspsncmp 20957 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 38364	If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 38363. TODO: can lspsncmp 20957 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 38365	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 38366	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 38367	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 38368*	Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 39166 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 38369*	Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 20969. (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 38370	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31374 analog.) Explicit atom version of lsmcv 20982. (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 38371*	The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 32080 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 38372*	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 32083 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 38373*	Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 32085 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 38374*	Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 32086 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 38375*	The ordering of two subspaces is determined by the atoms under them. (chrelat3 32093 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 38376*	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 32085 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 38377*	Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 38340. (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 38378	Extend class notation with the covering relation for a left module or left vector space.
class ⋖L
Definition	df-lcv 38379*	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 38381 for binary relation. (df-cv 32001 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 38380*	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 38381*	The covers relation for a left vector space (or a left module). (cvbr 32004 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 38382*	The covers relation for a left vector space (or a left module). (cvbr2 32005 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 38383*	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 38384	The covers relation implies proper subset. (cvpss 32007 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 38385	The covers relation implies no in-betweenness. (cvnbtwn 32008 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 38386	The covers relation is not transitive. (cvntr 32014 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 38387	The covers relation implies no in-betweenness. (cvnbtwn2 32009 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 38388	The covers relation implies no in-betweenness. (cvnbtwn3 32010 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 38389	Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 32015 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 38390*	If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 32088 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 38391	An atom covers the zero subspace. (atcv0 32064 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 38392	A subspace covered by an atom must be the zero subspace. (atcveq0 32070 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 38393	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 38394	Lemma for lcvexch 38399. (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 38395	Lemma for lcvexch 38399. (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 38396	Lemma for lcvexch 38399. (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 38397	Lemma for lcvexch 38399. (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 38398	Lemma for lcvexch 38399. (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 38399	Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 32091 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 38400	Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 32097 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)))