P. 349 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34801-34900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wl-lem-exsb 34801*	This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	wl-lem-nexmo 34802	This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑧))
Theorem	wl-lem-moexsb 34803*	The antecedent ∀𝑥(𝜑 → 𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑. This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))
Theorem	wl-alanbii 34804	This theorem extends alanimi 1813 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
Theorem	wl-mo2df 34805	Version of mof 2643 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))
Theorem	wl-mo2tf 34806	Closed form of mof 2643 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	wl-eudf 34807	Version of eu6 2655 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)))
Theorem	wl-eutf 34808	Closed form of eu6 2655 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
Theorem	wl-euequf 34809	euequ 2679 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
Theorem	wl-mo2t 34810*	Closed form of mof 2643. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	wl-mo3t 34811*	Closed form of mo3 2644. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	wl-sb8eut 34812	Substitution of variable in universal quantifier. Closed form of sb8eu 2682. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8mot 34813	Substitution of variable in universal quantifier. Closed form of sb8mo 2683. This theorem relates to wl-mo3t 34811, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2266 and sbco 2545. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 34811 in a simple fashion, unfortunately only if 𝑥 and 𝑦 are known to be distinct. To avoid any hassle with distinctors, we prefer to derive this theorem independently, ignoring the close connection between both theorems. From an educational standpoint, one would assume wl-mo3t 34811 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-axc11rc11 34814	Proving axc11r 2382 from axc11 2448. The hypotheses are two instances of axc11 2448 used in the proof here. Some systems introduce axc11 2448 as an axiom, see for example System S2 in https://us.metamath.org/downloads/finiteaxiom.pdf . By contrast, this database sees the variant axc11r 2382, directly derived from ax-12 2173, as foundational. Later axc11 2448 is proven somewhat trickily, requiring ax-10 2141 and ax-13 2386, see its proof. (Contributed by Wolf Lammen, 18-Jul-2023.)
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))    &   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Axiom	ax-wl-11v 34815*	Version of ax-11 2157 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2157. It will later be converted into a theorem directly based on ax-11 2157. (Contributed by Wolf Lammen, 28-Jun-2019.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	wl-ax11-lem1 34816	A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))
Theorem	wl-ax11-lem2 34817*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
Theorem	wl-ax11-lem3 34818*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)
Theorem	wl-ax11-lem4 34819*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-ax11-lem5 34820	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
Theorem	wl-ax11-lem6 34821*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦𝜑))
Theorem	wl-ax11-lem7 34822	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))
Theorem	wl-ax11-lem8 34823*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢∀𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥𝜑))
Theorem	wl-ax11-lem9 34824	The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑))
Theorem	wl-ax11-lem10 34825*	We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 194 can be used in conjunction with wl-ax11-lem9 34824 to eliminate the second antecedent. Missing is something along the lines of ax-6 1966, 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 → (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)))
Theorem	wl-clabv 34826*	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. 𝑥 ∈ 𝐴 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 𝑥 ∈ 𝐴 with 𝑥, 𝐴 not disjoint, is now only introduced either via ax-8 2112, ax-9 2120, or df-clel 2893. Theorem cleljust 2119 shows that a possible choice does not matter. The original df-clab 2800 can be rederived, see wl-dfclab 34827. 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 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	wl-dfclab 34827	Rederive df-clab 2800 from wl-clabv 34826. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	wl-clabtv 34828*	Using class abstraction in a context, requiring 𝑥 and 𝜑 disjoint, but based on fewer axioms than wl-clabt 34829. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})
Theorem	wl-clabt 34829	Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 34828. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)})
20.18.5  1. Restricted Quantifiers
Syntax	wl-ral 34830	Redefine the restricted universal quantifier context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from ∀𝑥 ∈ 𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∀(𝑥 : 𝐴)𝜑
Syntax	wl-rex 34831	Redefine the restricted existential quantifier context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from ∃𝑥 ∈ 𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∃(𝑥 : 𝐴)𝜑
Syntax	wl-rmo 34832	Redefine the restricted "at most one" quantifier context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from ∃*𝑥 ∈ 𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∃*(𝑥 : 𝐴)𝜑
Syntax	wl-reu 34833	Redefine the restricted existential uniqueness context to avoid overloading and syntax check errors in mmj2. This operator is to be distinguished from ∃!𝑥 ∈ 𝐴𝜑 with a similiar semantics, but using 𝑥 as a formal parameter rather than assuming true elementhood.
wff ∃!(𝑥 : 𝐴)𝜑
Syntax	wl-crab 34834	Redefine extended class notation to include the restricted class abstraction (class builder).
class {𝑥 : 𝐴 ∣ 𝜑}
Definition	df-wl-ral 34835*	The definiens of df-ral 3143, ∀𝑥(𝑥 ∈ 𝐴 → 𝜑) 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 = 𝑦 → 𝜑) is [𝑦 / 𝑥]𝜑 in disguise (see wl-dfralsb 34836). If 𝑥 is not free in 𝐴, a simplification is possible ( see wl-dfralf 34838, wl-dfralv 34840). (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(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	wl-dfralsb 34836*	An alternate definition of restricted universal quantification (df-wl-ral 34835) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑))
Theorem	wl-dfralflem 34837*	Lemma for wl-dfralf 34838 and wl-dfralv . (Contributed by Wolf Lammen, 23-May-2023.)
⊢ (∀𝑦∀𝑥(𝑦 ∈ 𝐴 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	wl-dfralf 34838	Restricted universal quantification (df-wl-ral 34835) allows a simplification, if we can assume all appearences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 23-May-2023.)
⊢ (Ⅎ𝑥𝐴 → (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
Theorem	wl-dfralfi 34839	Restricted universal quantification (df-wl-ral 34835) allows allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	wl-dfralv 34840*	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(𝑥 ∈ 𝐴 → 𝜑))
Theorem	wl-rgen 34841*	Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ∀(𝑥 : 𝐴)𝜑
Theorem	wl-rgenw 34842	Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.)
⊢ 𝜑    ⇒   ⊢ ∀(𝑥 : 𝐴)𝜑
Theorem	wl-rgen2w 34843	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
Theorem	wl-ralel 34844*	All elements of a class are elements of the class. (Contributed by Wolf Lammen, 10-Jun-2023.)
⊢ ∀(𝑥 : 𝐴)𝑥 ∈ 𝐴
Definition	df-wl-rex 34845	Restrict an existential quantifier to a class 𝐴. This version does not interpret elementhood verbatim as ∃𝑥 ∈ 𝐴𝜑 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 34835, where this is ruled out. Other definitions are wl-dfrexsb 34850 and wl-dfrexex 34849. If 𝑥 is not free in 𝐴, the defining expression can be simplified (see wl-dfrexf 34846, wl-dfrexv 34848). 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)
Theorem	wl-dfrexf 34846	Restricted existential quantification (df-wl-rex 34845) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 25-May-2023.)
⊢ (Ⅎ𝑥𝐴 → (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	wl-dfrexfi 34847	Restricted universal quantification (df-wl-rex 34845) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 26-May-2023.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	wl-dfrexv 34848*	Alternate definition of restricted universal quantification (df-wl-rex 34845) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 25-May-2023.)
⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	wl-dfrexex 34849*	Alternate definition of the restricted existential quantification (df-wl-rex 34845), according to the pattern given in df-wl-ral 34835. (Contributed by Wolf Lammen, 25-May-2023.)
⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	wl-dfrexsb 34850*	An alternate definition of restricted existential quantification (df-wl-rex 34845) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
⊢ (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
Definition	df-wl-rmo 34851*	Restrict "at most one" to a given class 𝐴. This version does not interpret elementhood verbatim like ∃*𝑥 ∈ 𝐴𝜑 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 34835, 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 34852 and, if 𝑥 is not free in 𝐴, wl-dfrmov 34853 and wl-dfrmof 34854. (Contributed by NM, 30-Aug-1993.) Isolate x from A. (Revised by Wolf Lammen, 26-May-2023.)
⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃𝑦∀(𝑥 : 𝐴)(𝜑 → 𝑥 = 𝑦))
Theorem	wl-dfrmosb 34852*	An alternate definition of restricted "at most one" (df-wl-rmo 34851) using substitution. (Contributed by Wolf Lammen, 27-May-2023.)
⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
Theorem	wl-dfrmov 34853*	Alternate definition of restricted "at most one" (df-wl-rmo 34851) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
⊢ (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	wl-dfrmof 34854	Restricted "at most one" (df-wl-rmo 34851) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)
⊢ (Ⅎ𝑥𝐴 → (∃*(𝑥 : 𝐴)𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
Definition	df-wl-reu 34855	Restrict existential uniqueness to a given class 𝐴. This version does not interpret elementhood verbatim like ∃!𝑥 ∈ 𝐴𝜑 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 34835, 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 34856 and, if 𝑥 is not free in 𝐴, wl-dfreuv 34857 and wl-dfreuf 34858. (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))
Theorem	wl-dfreusb 34856*	An alternate definition of restricted existential uniqueness (df-wl-reu 34855) using substitution. (Contributed by Wolf Lammen, 28-May-2023.)
⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
Theorem	wl-dfreuv 34857*	Alternate definition of restricted existential uniqueness (df-wl-reu 34855) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 28-May-2023.)
⊢ (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	wl-dfreuf 34858	Restricted existential uniqueness (df-wl-reu 34855) allows a simplification, if we can assume all occurrences of 𝑥 in 𝐴 are bounded. (Contributed by Wolf Lammen, 28-May-2023.)
⊢ (Ⅎ𝑥𝐴 → (∃!(𝑥 : 𝐴)𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
Definition	df-wl-rab 34859*	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 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))}
Theorem	wl-dfrabsb 34860*	Alternate definition of restricted class abstraction (df-wl-rab 34859), using substitution. (Contributed by Wolf Lammen, 28-May-2023.)
⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)}
Theorem	wl-dfrabv 34861*	Alternate definition of restricted class abstraction (df-wl-rab 34859), when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ {𝑥 : 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	wl-clelsb3df 34862	Deduction version of clelsb3f 2982. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝐴)    ⇒   ⊢ (𝜑 → ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
Theorem	wl-dfrabf 34863	Alternate definition of restricted class abstraction (df-wl-rab 34859), when 𝑥 is not free in 𝐴. (Contributed by Wolf Lammen, 29-May-2023.)
⊢ (Ⅎ𝑥𝐴 → {𝑥 : 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
20.19  Mathbox for Brendan Leahy
Theorem	rabiun 34864*	Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
⊢ {𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑} = ∪ 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝜑}
Theorem	iundif1 34865*	Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶)
Theorem	imadifss 34866	The difference of images is a subset of the image of the difference. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))
Theorem	cureq 34867	Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Theorem	unceq 34868	Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
Theorem	curf 34869	Functional property of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))
Theorem	uncf 34870	Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
Theorem	curfv 34871	Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))
Theorem	uncov 34872	Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))
Theorem	curunc 34873	Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
Theorem	unccur 34874	Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)
⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)
Theorem	phpreu 34875*	Theorem related to pigeonhole principle. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝑥 = 𝐶))
Theorem	finixpnum 34876*	A finite Cartesian product of numerable sets is numerable. (Contributed by Brendan Leahy, 24-Feb-2019.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)
Theorem	fin2solem 34877*	Lemma for fin2so 34878. (Contributed by Brendan Leahy, 29-Jun-2019.)
⊢ ((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → (𝑦𝑅𝑧 → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧}))
Theorem	fin2so 34878	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 ∈ FinII ∧ 𝑅 Or 𝐴) → 𝐴 ∈ Fin)
Theorem	ltflcei 34879	Theorem to move the floor function across a strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘𝐴) < 𝐵 ↔ 𝐴 < -(⌊‘-𝐵)))
Theorem	leceifl 34880	Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-(⌊‘-𝐴) ≤ 𝐵 ↔ 𝐴 ≤ (⌊‘𝐵)))
Theorem	sin2h 34881	Half-angle rule for sine. (Contributed by Brendan Leahy, 3-Aug-2018.)
⊢ (𝐴 ∈ (0[,](2 · π)) → (sin‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / 2)))
Theorem	cos2h 34882	Half-angle rule for cosine. (Contributed by Brendan Leahy, 4-Aug-2018.)
⊢ (𝐴 ∈ (-π[,]π) → (cos‘(𝐴 / 2)) = (√‘((1 + (cos‘𝐴)) / 2)))
Theorem	tan2h 34883	Half-angle rule for tangent. (Contributed by Brendan Leahy, 4-Aug-2018.)
⊢ (𝐴 ∈ (0[,)π) → (tan‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / (1 + (cos‘𝐴)))))
Theorem	lindsadd 34884	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))
Theorem	lindsdom 34885	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))) → 𝑋 ≼ 𝐼)
Theorem	lindsenlbs 34886	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))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))
Theorem	matunitlindflem1 34887	One direction of matunitlindf 34889. (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) = (0g‘𝑅)))
Theorem	matunitlindflem2 34888	One direction of matunitlindf 34889. (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))
Theorem	matunitlindf 34889	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)))
Theorem	ptrest 34890*	Expressing a restriction of a product topology as a product topology. (Contributed by Brendan Leahy, 24-Mar-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Top)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((∏t‘𝐹) ↾t X𝑘 ∈ 𝐴 𝑆) = (∏t‘(𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) ↾t 𝑆))))
Theorem	ptrecube 34891*	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))    ⇒   ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
Theorem	poimirlem1 34892*	Lemma for poimir 34924- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ)    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    ⇒   ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))
Theorem	poimirlem2 34893*	Lemma for poimir 34924- consecutive vertices differ in at most one dimension. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ)    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ (𝜑 → 𝑉 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 ∈ ((0...𝑁) ∖ {𝑉}))    ⇒   ⊢ (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹‘𝑉)‘𝑛))
Theorem	poimirlem3 34894*	Lemma for poimir 34924 to add an interior point to an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    &   ⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾))    &   ⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀))    ⇒   ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(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})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
Theorem	poimirlem4 34895*	Lemma for poimir 34924 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.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    ⇒   ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
Theorem	poimirlem5 34896*	Lemma for poimir 34924 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.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 0 < (2nd ‘𝑇))    ⇒   ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))
Theorem	poimirlem6 34897*	Lemma for poimir 34924 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.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 ∈ (1...((2nd ‘𝑇) − 1)))    ⇒   ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑀))
Theorem	poimirlem7 34898*	Lemma for poimir 34924, similar to poimirlem6 34897, but for vertices after the opposite vertex. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))    ⇒   ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑀))
Theorem	poimirlem8 34899*	Lemma for poimir 34924, establishing that away from the opposite vertex the walks in poimirlem9 34900 yield the same vertices. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
Theorem	poimirlem9 34900*	Lemma for poimir 34924, 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.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ≠ (2nd ‘(1st ‘𝑇)))    ⇒   ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) = ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))))