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Theorem List for Metamath Proof Explorer - 34701-34800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwl-ax11-lem3 34701* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
 
Theoremwl-ax11-lem4 34702* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
 
Theoremwl-ax11-lem5 34703 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
 
Theoremwl-ax11-lem6 34704* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))
 
Theoremwl-ax11-lem7 34705 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))
 
Theoremwl-ax11-lem8 34706* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))
 
Theoremwl-ax11-lem9 34707 The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))
 
Theoremwl-ax11-lem10 34708* We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 193 can be used in conjunction with wl-ax11-lem9 34707 to eliminate the second antecedent. Missing is something along the lines of ax-6 1961, so we could remove the first antecedent. But the Metamath axioms cannot accomplish this. Such a rule must reside one abstraction level higher than all others: It says that a distinctor implies a distinct variable condition on its contained setvar. This is only needed if such conditions are required, as ax-11v does. The result of this study is for me, that you cannot introduce a setvar capturing this condition, and hope to eliminate it later. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑦 𝑦 = 𝑢 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)))
 
Theoremwl-clabv 34709* Variant of df-clab 2800, where the element 𝑥 is required to be disjoint from the class it is taken from. This restriction meets similar ones found in other definitions and axioms like ax-ext 2793, df-clel 2893 and df-cleq 2814. 𝑥𝐴 with 𝐴 depending on 𝑥 can be the source of side effects, that you rather want to be aware of. So here we eliminate one possible way of letting this slip in instead.

An expression 𝑥𝐴 with 𝑥, 𝐴 not disjoint, is now only introduced either via ax-8 2107, ax-9 2115, or df-clel 2893. Theorem cleljust 2114 shows that a possible choice does not matter.

The original df-clab 2800 can be rederived, see wl-dfclab 34710. In an implementation this theorem is the only user of df-clab. (Contributed by NM, 26-May-1993.) Element and class are disjoint. (Revised by Wolf Lammen, 31-May-2023.)

(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
 
Theoremwl-dfclab 34710 Rederive df-clab 2800 from wl-clabv 34709. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.)
(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
 
Theoremwl-clabtv 34711* Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 34712. (Contributed by Wolf Lammen, 29-May-2023.)
(𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})
 
Theoremwl-clabt 34712 Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 34711. (Contributed by Wolf Lammen, 29-May-2023.)
𝑥𝜑       (𝜑 → {𝑥𝜓} = {𝑥 ∣ (𝜑𝜓)})
 
20.18.5  1. Restricted Quantifiers
 
Syntaxwl-ral 34713 Redefine the restricted universal quantifier context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from 𝑥𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∀(𝑥 : 𝐴)𝜑
 
Syntaxwl-rex 34714 Redefine the restricted existential quantifier context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from 𝑥𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∃(𝑥 : 𝐴)𝜑
 
Syntaxwl-rmo 34715 Redefine the restricted "at most one" quantifier context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from ∃*𝑥𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∃*(𝑥 : 𝐴)𝜑
 
Syntaxwl-reu 34716 Redefine the restricted existential uniqueness context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from ∃!𝑥𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∃!(𝑥 : 𝐴)𝜑
 
Syntaxwl-crab 34717 Redefine extended class notation to include the restricted class abstraction (class builder).
class {𝑥 : 𝐴𝜑}
 
Definitiondf-wl-ral 34718* The definiens of df-ral 3143, 𝑥(𝑥𝐴𝜑) is a short and simple expression, but has a severe downside: It allows for two substantially different interpretations. One, and this is the common case, wants this expression to denote that 𝜑 holds for all elements of 𝐴. To this end, 𝑥 must not be free in 𝐴, though . Should instead 𝐴 vary with 𝑥, then we rather focus on those 𝑥, that happen to be an element of their respective 𝐴(𝑥). Such interpretation is rare, but must nevertheless be considered in design and comments.

In addition, many want definitions be designed to express just a single idea, not many.

Our definition here introduces a dummy variable 𝑦, disjoint from all other variables, to describe an element in 𝐴. It lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurrences in 𝜑 are still honored.

The resulting subexpression 𝑥(𝑥 = 𝑦𝜑) is [𝑦 / 𝑥]𝜑 in disguise (see wl-dfralsb 34719).

If 𝑥 is not free in 𝐴, a simplification is possible ( see wl-dfralf 34721, wl-dfralv 34723). (Contributed by NM, 19-Aug-1993.) Isolate x from A, idea of Mario Carneiro. (Revised by Wolf Lammen, 21-May-2023.)

(∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremwl-dfralsb 34719* An alternate definition of restricted universal quantification (df-wl-ral 34718) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
(∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑))
 
Theoremwl-dfralflem 34720* Lemma for wl-dfralf 34721 and wl-dfralv . (Contributed by Wolf Lammen, 23-May-2023.)
(∀𝑦𝑥(𝑦𝐴 → (𝑥 = 𝑦𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremwl-dfralf 34721 Restricted universal quantification (df-wl-ral 34718) allows a simplification, if we can assume all appearences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 23-May-2023.)
(𝑥𝐴 → (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑)))
 
Theoremwl-dfralfi 34722 Restricted universal quantification (df-wl-ral 34718) allows allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
𝑥𝐴       (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremwl-dfralv 34723* Alternate definition of restricted universal quantification (df-wl-ral ) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 23-May-2023.)
(∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
 
Theoremwl-rgen 34724* Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.)
(𝑥𝐴𝜑)       ∀(𝑥 : 𝐴)𝜑
 
Theoremwl-rgenw 34725 Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.)
𝜑       ∀(𝑥 : 𝐴)𝜑
 
Theoremwl-rgen2w 34726 Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by Wolf Lammen, 10-Jun-2023.)
𝜑       ∀(𝑥 : 𝐴)∀(𝑦 : 𝐵)𝜑
 
Theoremwl-ralel 34727* All elements of a class are elements of the class. (Contributed by Wolf Lammen, 10-Jun-2023.)
∀(𝑥 : 𝐴)𝑥𝐴
 
Definitiondf-wl-rex 34728 Restrict an existential quantifier to a class 𝐴. This version does not interpret elementhood verbatim as 𝑥𝐴𝜑 does. Assuming a real elementhood can lead to awkward consequences should the class 𝐴 depend on 𝑥. Instead we base the definition on df-wl-ral 34718, where this is ruled out. Other definitions are wl-dfrexsb 34733 and wl-dfrexex 34732. If 𝑥 is not free in 𝐴, the defining expression can be simplified (see wl-dfrexf 34729, wl-dfrexv 34731).

This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurrences in 𝜑 are fully honored. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 25-May-2023.)

(∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
 
Theoremwl-dfrexf 34729 Restricted existential quantification (df-wl-rex 34728) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 25-May-2023.)
(𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥𝐴𝜑)))
 
Theoremwl-dfrexfi 34730 Restricted universal quantification (df-wl-rex 34728) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
𝑥𝐴       (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Theoremwl-dfrexv 34731* Alternate definition of restricted universal quantification (df-wl-rex 34728) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 25-May-2023.)
(∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
 
Theoremwl-dfrexex 34732* Alternate definition of the restricted existential quantification (df-wl-rex 34728), according to the pattern given in df-wl-ral 34718. (Contributed by Wolf Lammen, 25-May-2023.)
(∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 
Theoremwl-dfrexsb 34733* An alternate definition of restricted existential quantification (df-wl-rex 34728) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
(∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
 
Definitiondf-wl-rmo 34734* Restrict "at most one" to a given class 𝐴. This version does not interpret elementhood verbatim like ∃*𝑥𝐴𝜑 does. Assuming a real elementhood can lead to awkward consequences should the class 𝐴 depend on 𝑥. Instead we base the definition on df-wl-ral 34718, where this is already ruled out.

This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurrences in 𝜑 are fully honored.

Alternate definitions are given in wl-dfrmosb 34735 and, if 𝑥 is not free in 𝐴, wl-dfrmov 34736 and wl-dfrmof 34737. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 26-May-2023.)

(∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑𝑥 = 𝑦))
 
Theoremwl-dfrmosb 34735* An alternate definition of restricted "at most one" (df-wl-rmo 34734) using substitution. (Contributed by Wolf Lammen, 27-May-2023.)
(∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
 
Theoremwl-dfrmov 34736* Alternate definition of restricted "at most one" (df-wl-rmo 34734) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
(∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
 
Theoremwl-dfrmof 34737 Restricted "at most one" (df-wl-rmo 34734) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)
(𝑥𝐴 → (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑)))
 
Definitiondf-wl-reu 34738 Restrict existential uniqueness to a given class 𝐴. This version does not interpret elementhood verbatim like ∃!𝑥𝐴𝜑 does. Assuming a real elementhood can lead to awkward consequences should the class 𝐴 depend on 𝑥. Instead we base the definition on df-wl-ral 34718, where this is ruled out.

This definition lets 𝑥 appear as a formal parameter with no connection to 𝐴, but occurrences in 𝜑 are fully honored.

Alternate definitions are given in wl-dfreusb 34739 and, if 𝑥 is not free in 𝐴, wl-dfreuv 34740 and wl-dfreuf 34741. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 28-May-2023.)

(∃!(𝑥 : 𝐴)𝜑 ↔ (∃(𝑥 : 𝐴)𝜑 ∧ ∃*(𝑥 : 𝐴)𝜑))
 
Theoremwl-dfreusb 34739* An alternate definition of restricted existential uniqueness (df-wl-reu 34738) using substitution. (Contributed by Wolf Lammen, 28-May-2023.)
(∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
 
Theoremwl-dfreuv 34740* Alternate definition of restricted existential uniqueness (df-wl-reu 34738) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
(∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
 
Theoremwl-dfreuf 34741 Restricted existential uniqueness (df-wl-reu 34738) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)
(𝑥𝐴 → (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑)))
 
Definitiondf-wl-rab 34742* Define a restricted class abstraction (class builder), which is the class of all 𝑥 in 𝐴 such that 𝜑 is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.) Isolate x from A. (Revised by Wolf Lammen, 28-May-2023.)
{𝑥 : 𝐴𝜑} = {𝑦 ∣ (𝑦𝐴 ∧ ∀𝑥(𝑥 = 𝑦𝜑))}
 
Theoremwl-dfrabsb 34743* Alternate definition of restricted class abstraction (df-wl-rab 34742), using substitution. (Contributed by Wolf Lammen, 28-May-2023.)
{𝑥 : 𝐴𝜑} = {𝑦 ∣ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)}
 
Theoremwl-dfrabv 34744* Alternate definition of restricted class abstraction (df-wl-rab 34742), when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 29-May-2023.)
{𝑥 : 𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
 
Theoremwl-clelsb3df 34745 Deduction version of clelsb3f 2982. (Contributed by Wolf Lammen, 29-May-2023.)
𝑦𝜑    &   (𝜑𝑦𝐴)       (𝜑 → ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴))
 
Theoremwl-dfrabf 34746 Alternate definition of restricted class abstraction (df-wl-rab 34742), when 𝑥 is not free in 𝐴. (Contributed by Wolf Lammen, 29-May-2023.)
(𝑥𝐴 → {𝑥 : 𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)})
 
20.19  Mathbox for Brendan Leahy
 
Theoremrabiun 34747* Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
{𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}
 
Theoremiundif1 34748* Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
 
Theoremimadifss 34749 The difference of images is a subset of the image of the difference. (Contributed by Brendan Leahy, 21-Aug-2020.)
((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
 
Theoremcureq 34750 Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
 
Theoremunceq 34751 Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
 
Theoremcurf 34752 Functional property of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶m 𝐵))
 
Theoremuncf 34753 Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
 
Theoremcurfv 34754 Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
 
Theoremuncov 34755 Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))
 
Theoremcurunc 34756 Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
 
Theoremunccur 34757 Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)
((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = 𝐹)
 
Theoremphpreu 34758* Theorem related to pigeonhole principle. (Contributed by Brendan Leahy, 21-Aug-2020.)
((𝐴 ∈ Fin ∧ 𝐴𝐵) → (∀𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥 = 𝐶))
 
Theoremfinixpnum 34759* A finite Cartesian product of numerable sets is numerable. (Contributed by Brendan Leahy, 24-Feb-2019.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ dom card) → X𝑥𝐴 𝐵 ∈ dom card)
 
Theoremfin2solem 34760* Lemma for fin2so 34761. (Contributed by Brendan Leahy, 29-Jun-2019.)
((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))
 
Theoremfin2so 34761 Any totally ordered Tarski-finite set is finite; in particular, no amorphous set can be ordered. Theorem 2 of [Levy58]] p. 4. (Contributed by Brendan Leahy, 28-Jun-2019.)
((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝐴 ∈ Fin)
 
Theoremltflcei 34762 Theorem to move the floor function across a strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘𝐴) < 𝐵𝐴 < -(⌊‘-𝐵)))
 
Theoremleceifl 34763 Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-(⌊‘-𝐴) ≤ 𝐵𝐴 ≤ (⌊‘𝐵)))
 
Theoremsin2h 34764 Half-angle rule for sine. (Contributed by Brendan Leahy, 3-Aug-2018.)
(𝐴 ∈ (0[,](2 · π)) → (sin‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / 2)))
 
Theoremcos2h 34765 Half-angle rule for cosine. (Contributed by Brendan Leahy, 4-Aug-2018.)
(𝐴 ∈ (-π[,]π) → (cos‘(𝐴 / 2)) = (√‘((1 + (cos‘𝐴)) / 2)))
 
Theoremtan2h 34766 Half-angle rule for tangent. (Contributed by Brendan Leahy, 4-Aug-2018.)
(𝐴 ∈ (0[,)π) → (tan‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / (1 + (cos‘𝐴)))))
 
Theoremlindsadd 34767 In a vector space, the union of an independent set and a vector not in its span is an independent set. (Contributed by Brendan Leahy, 4-Mar-2023.)
((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝑋 ∈ ((Base‘𝑊) ∖ ((LSpan‘𝑊)‘𝐹))) → (𝐹 ∪ {𝑋}) ∈ (LIndS‘𝑊))
 
Theoremlindsdom 34768 A linearly independent set in a free linear module of finite dimension over a division ring is smaller than the dimension of the module. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) → 𝑋𝐼)
 
Theoremlindsenlbs 34769 A maximal linearly independent set in a free module of finite dimension over a division ring is a basis. (Contributed by Brendan Leahy, 2-Jun-2021.)
(((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))
 
Theoremmatunitlindflem1 34770 One direction of matunitlindf 34772. (Contributed by Brendan Leahy, 2-Jun-2021.)
(((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g𝑅)))
 
Theoremmatunitlindflem2 34771 One direction of matunitlindf 34772. (Contributed by Brendan Leahy, 2-Jun-2021.)
((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
 
Theoremmatunitlindf 34772 A matrix over a field is invertible iff the rows are linearly independent. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Unit‘(𝐼 Mat 𝑅)) ↔ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)))
 
Theoremptrest 34773* Expressing a restriction of a product topology as a product topology. (Contributed by Brendan Leahy, 24-Mar-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   ((𝜑𝑘𝐴) → 𝑆𝑊)       (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
 
Theoremptrecube 34774* Any point in an open set of N-space is surrounded by an open cube within that set. (Contributed by Brendan Leahy, 21-Aug-2020.) (Proof shortened by AV, 28-Sep-2020.)
𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       ((𝑆𝑅𝑃𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
 
Theorempoimirlem1 34775* Lemma for poimir 34807- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))    &   (𝜑𝑇:(1...𝑁)⟶ℤ)    &   (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))       (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
 
Theorempoimirlem2 34776* Lemma for poimir 34807- consecutive vertices differ in at most one dimension. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))    &   (𝜑𝑇:(1...𝑁)⟶ℤ)    &   (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   (𝜑𝑉 ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 ∈ ((0...𝑁) ∖ {𝑉}))       (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛))
 
Theorempoimirlem3 34777* Lemma for poimir 34807 to add an interior point to an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑀 < 𝑁)    &   (𝜑𝑇:(1...𝑀)⟶(0..^𝐾))    &   (𝜑𝑈:(1...𝑀)–1-1-onto→(1...𝑀))       (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
 
Theorempoimirlem4 34778* Lemma for poimir 34807 connecting the admissible faces on the back face of the (𝑀 + 1)-cube to admissible simplices in the 𝑀-cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑀 < 𝑁)       (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
 
Theorempoimirlem5 34779* Lemma for poimir 34807 to establish that, for the simplices defined by a walk along the edges of an 𝑁-cube, if the starting vertex is not opposite a given face, it is the earliest vertex of the face on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝑇𝑆)    &   (𝜑 → 0 < (2nd𝑇))       (𝜑 → (𝐹‘0) = (1st ‘(1st𝑇)))
 
Theorempoimirlem6 34780* Lemma for poimir 34807 establishing, for a face of a simplex defined by a walk along the edges of an 𝑁-cube, the single dimension in which successive vertices before the opposite vertex differ. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 ∈ (1...((2nd𝑇) − 1)))       (𝜑 → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑀))
 
Theorempoimirlem7 34781* Lemma for poimir 34807, similar to poimirlem6 34780, but for vertices after the opposite vertex. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 ∈ ((((2nd𝑇) + 1) + 1)...𝑁))       (𝜑 → (𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st𝑇))‘𝑀))
 
Theorempoimirlem8 34782* Lemma for poimir 34807, establishing that away from the opposite vertex the walks in poimirlem9 34783 yield the same vertices. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))    &   (𝜑𝑈𝑆)       (𝜑 → ((2nd ‘(1st𝑈)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((2nd ‘(1st𝑇)) ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
 
Theorempoimirlem9 34783* Lemma for poimir 34807, establishing the two walks that yield a given face when the opposite vertex is neither first nor last. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))    &   (𝜑𝑈𝑆)    &   (𝜑 → (2nd ‘(1st𝑈)) ≠ (2nd ‘(1st𝑇)))       (𝜑 → (2nd ‘(1st𝑈)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
 
Theorempoimirlem10 34784* Lemma for poimir 34807 establishing the cube that yields the simplex that yields a face if the opposite vertex was first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) = 0)       (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
 
Theorempoimirlem11 34785* Lemma for poimir 34807 connecting walks that could yield from a given cube a given face opposite the first vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) = 0)    &   (𝜑𝑈𝑆)    &   (𝜑 → (2nd𝑈) = 0)    &   (𝜑𝑀 ∈ (1...𝑁))       (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st𝑈)) “ (1...𝑀)))
 
Theorempoimirlem12 34786* Lemma for poimir 34807 connecting walks that could yield from a given cube a given face opposite the final vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) = 𝑁)    &   (𝜑𝑈𝑆)    &   (𝜑 → (2nd𝑈) = 𝑁)    &   (𝜑𝑀 ∈ (0...(𝑁 − 1)))       (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st𝑈)) “ (1...𝑀)))
 
Theorempoimirlem13 34787* Lemma for poimir 34807- for at most one simplex associated with a shared face is the opposite vertex first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))       (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
 
Theorempoimirlem14 34788* Lemma for poimir 34807- for at most one simplex associated with a shared face is the opposite vertex last on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))       (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
 
Theorempoimirlem15 34789* Lemma for poimir 34807, that the face in poimirlem22 34796 is a face. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))       (𝜑 → ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ 𝑆)
 
Theorempoimirlem16 34790* Lemma for poimir 34807 establishing the vertices of the simplex of poimirlem17 34791. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)    &   (𝜑 → (2nd𝑇) = 0)       (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
 
Theorempoimirlem17 34791* Lemma for poimir 34807 establishing existence for poimirlem18 34792. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)    &   (𝜑 → (2nd𝑇) = 0)       (𝜑 → ∃𝑧𝑆 𝑧𝑇)
 
Theorempoimirlem18 34792* Lemma for poimir 34807 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)    &   (𝜑 → (2nd𝑇) = 0)       (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
 
Theorempoimirlem19 34793* Lemma for poimir 34807 establishing the vertices of the simplex in poimirlem20 34794. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)    &   (𝜑 → (2nd𝑇) = 𝑁)       (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
 
Theorempoimirlem20 34794* Lemma for poimir 34807 establishing existence for poimirlem21 34795. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)    &   (𝜑 → (2nd𝑇) = 𝑁)       (𝜑 → ∃𝑧𝑆 𝑧𝑇)
 
Theorempoimirlem21 34795* Lemma for poimir 34807 stating that, given a face not on a back face of the cube and a simplex in which it's opposite the final point of the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)    &   (𝜑 → (2nd𝑇) = 𝑁)       (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
 
Theorempoimirlem22 34796* Lemma for poimir 34807, that a given face belongs to exactly two simplices, provided it's not on the boundary of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   (𝜑𝑇𝑆)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)    &   ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)       (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
 
Theorempoimirlem23 34797* Lemma for poimir 34807, two ways of expressing the property that a face is not on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶(0..^𝐾))    &   (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   (𝜑𝑉 ∈ (0...𝑁))       (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁))))
 
Theorempoimirlem24 34798* Lemma for poimir 34807, two ways of expressing that a simplex has an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    &   (𝜑𝑇:(1...𝑁)⟶(0..^𝐾))    &   (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   (𝜑𝑉 ∈ (0...𝑁))       (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = 𝑇, 𝑈⟩ / 𝑠𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁)))))
 
Theorempoimirlem25 34799* Lemma for poimir 34807 stating that for a given simplex such that no vertex maps to 𝑁, the number of admissible faces is even. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    &   (𝜑𝑇:(1...𝑁)⟶(0..^𝐾))    &   (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ((𝜑𝑗 ∈ (0...𝑁)) → 𝑁𝑇, 𝑈⟩ / 𝑠𝐶)       (𝜑 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = 𝑇, 𝑈⟩ / 𝑠𝐶}))
 
Theorempoimirlem26 34800* Lemma for poimir 34807 showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))       (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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