P. 379 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37801-37900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dveeq1-o16 37801*	Version of dveeq1 2379 using ax-c16 37757 instead of ax-5 1913. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1913. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax5el 37802*	Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1913 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)
Theorem	axc11n-16 37803*	This theorem shows that, given ax-c16 37757, we can derive a version of ax-c11n 37753. However, it is weaker than ax-c11n 37753 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Theorem	dveel2ALT 37804*	Alternate proof of dveel2 2461 using ax-c16 37757 instead of ax-5 1913. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
Theorem	ax12f 37805	Basis step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. We can start with any formula 𝜑 in which 𝑥 is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
Theorem	ax12eq 37806	Basis step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. 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 37807	Basis step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. 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 37808	Induction step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. 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 37809	Induction step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. 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 37810	Lemma for ax12inda2 37812 and ax12inda 37813. (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 37811*	Alternate proof of ax12inda2 37812, slightly more direct and not requiring ax-c16 37757. (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 37812*	Induction step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. Quantification case. When 𝑧 and 𝑦 are distinct, this theorem avoids the dummy variables needed by the more general ax12inda 37813. (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 37813*	Induction step for constructing a substitution instance of ax-c15 37754 without using ax-c15 37754. Quantification case. (When 𝑧 and 𝑦 are distinct, ax12inda2 37812 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 37814*	Rederivation of ax-c15 37754 from ax12v 2172 (without using ax-c15 37754 or the full ax-12 2171). Thus, the hypothesis (ax12v 2172) provides an alternate axiom that can be used in place of ax-c15 37754. See also axc15 2421. (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 37815*	Derive ax-c15 37754 from a hypothesis in the form of ax-12 2171, without using ax-12 2171 or ax-c15 37754. The hypothesis is weaker than ax-12 2171, 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 2171, if we also have ax-c11 37752, which this proof uses. As Theorem ax12 2422 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 37753 instead of ax-c11 37752. (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 37816	Show that ax-c11 37752 can be derived from ax-c11n 37753 and ax-12 2171. An open problem is whether this theorem can be derived from ax-c11n 37753 and the others when ax-12 2171 is replaced with ax-c15 37754 or ax12v 2172. See Theorems axc11nfromc11 37791 for the rederivation of ax-c11n 37753 from axc11 2429. Normally, axc11 2429 should be used rather than ax-c11 37752 or axc11-o 37816, 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 37817*	Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 15725. The proof demonstrates how this can be derived starting from from fsumshft 15725. (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 37818	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 6495. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) WARNING: THIS "AXIOM", WHICH IS THE OLD df-riota 7364, CONFLICTS WITH (THE NEW) df-riota 7364 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 37819*	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 37820*	Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
Theorem	riotasvd 37821*	Deduction version of riotasv 37824. (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 37822*	Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5401). Special case of riota2f 7389. (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 37823*	The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5401) in the form of a substitution instance. Special case of riota2f 7389. (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 37824*	Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5401). Special case of riota2f 7389. (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 37825*	A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5401) 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 37826	A version of elimhyp 4593 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
⊢ [𝐵 / 𝑥]𝜑    ⇒   ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
Theorem	dedths 37827	A version of weak deduction theorem dedth 4586 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	renegclALT 37828	Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 11522. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
Theorem	elimhyps2 37829	Generalization of elimhyps 37826 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 37830	Generalization of dedths 37827 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 37831	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 37832	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 3698. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2911. (Contributed by NM, 19-Nov-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶)    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐶)
Theorem	nfded2 37833	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 4890) that starts from abidnf 3698. The last is assigned to the inference form (e.g., Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4889) whose hypotheses are satisfied using nfaba1 2911. (Contributed by NM, 19-Nov-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    &   ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷)    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐷)
Theorem	nfunidALT2 37834	Deduction version of nfuni 4915. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	nfunidALT 37835	Deduction version of nfuni 4915. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	nfopdALT 37836	Deduction version of bound-variable hypothesis builder nfop 4889. This shows how the deduction version of a not-free theorem such as nfop 4889 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 37837	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 37838*	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 37839	Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
class LSAtoms
Syntax	clsh 37840	Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
class LSHyp
Definition	df-lsatoms 37841*	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 37842*	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 37843*	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 37844*	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 37845*	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 37846	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 37847	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 37848	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 37849	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 37850	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 37851	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 37852	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 37853	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 37854	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 37855*	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 37856*	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 37857	The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 37858 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 37858	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 37859*	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 37860*	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 37861	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 37862	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 37863	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 37864	A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 31593 analog.) (Contributed by NM, 25-Aug-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑈 ≠ { 0 })
Theorem	lsatspn0 37865	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 37866	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 37867	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 37868	If two atoms are comparable, they are equal. (atsseq 31595 analog.) TODO: can lspsncmp 20728 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 37869	If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 37868. TODO: can lspsncmp 20728 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 37870	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 37871	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 37872	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 37873*	Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 38671 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 37874*	Show equality with the span of the sum of two vectors, one of which (𝑋) is fixed in advance. Compare lspfixed 20740. (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 37875	Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 30900 analog.) Explicit atom version of lsmcv 20753. (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 37876*	The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 31606 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 37877*	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 31609 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 37878*	Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 31611 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 37879*	Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 31612 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 37880*	The ordering of two subspaces is determined by the atoms under them. (chrelat3 31619 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 37881*	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 31611 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 37882*	Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 37845. (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 37883	Extend class notation with the covering relation for a left module or left vector space.
class ⋖L
Definition	df-lcv 37884*	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 37886 for binary relation. (df-cv 31527 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 37885*	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 37886*	The covers relation for a left vector space (or a left module). (cvbr 31530 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 37887*	The covers relation for a left vector space (or a left module). (cvbr2 31531 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 37888*	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 37889	The covers relation implies proper subset. (cvpss 31533 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 37890	The covers relation implies no in-betweenness. (cvnbtwn 31534 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 37891	The covers relation is not transitive. (cvntr 31540 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 37892	The covers relation implies no in-betweenness. (cvnbtwn2 31535 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 37893	The covers relation implies no in-betweenness. (cvnbtwn3 31536 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 37894	Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 31541 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 37895*	If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 31614 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 37896	An atom covers the zero subspace. (atcv0 31590 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 37897	A subspace covered by an atom must be the zero subspace. (atcveq0 31596 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 37898	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 37899	Lemma for lcvexch 37904. (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 37900	Lemma for lcvexch 37904. (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)