P. 326 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wl-aleq 32501	The semantics of ∀𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
Theorem	wl-nfeqfb 32502	Extend nfeqf 2289 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
Theorem	wl-nfs1t 32503	If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2353. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	wl-equsald 32504	Deduction version of equsal 2279. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
Theorem	wl-equsal 32505	A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 32504 first, and then deriving more specialized versions wl-equsal 32505 and wl-equsal1t 32506 then is more efficient than the other way round, which is possible, too. See also equsal 2279. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	wl-equsal1t 32506	The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑. This theorem is more fundamental than equsal 2279, spimt 2241 or sbft 2367, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
Theorem	wl-equsalcom 32507	This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑦 = 𝑥 → 𝜑))
Theorem	wl-equsal1i 32508	The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)    &   ⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	wl-sb6rft 32509	A specialization of wl-equsal1t 32506. Closed form of sb6rf 2411. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))
Theorem	wl-sbrimt 32510	Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2384. (Contributed by Wolf Lammen, 26-Jul-2019.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
Theorem	wl-sblimt 32511	Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2384. (Contributed by Wolf Lammen, 26-Jul-2019.)
⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))
Theorem	wl-sb8t 32512	Substitution of variable in universal quantifier. Closed form of sb8 2412. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8et 32513	Substitution of variable in universal quantifier. Closed form of sb8e 2413. (Contributed by Wolf Lammen, 27-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sbhbt 32514	Closed form of sbhb 2426. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Theorem	wl-sbnf1 32515	Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2427. Note: This theorem shows that sbnf2 2427 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Theorem	wl-equsb3 32516	equsb3 2420 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
Theorem	wl-equsb4 32517	Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
Theorem	wl-sb6nae 32518	Version of sb6 2417 suitable for elimination of unnecessary dv restrictions. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	wl-sb5nae 32519	Version of sb5 2418 suitable for elimination of unnecessary dv restrictions. (Contributed by Wolf Lammen, 28-Jul-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	wl-2sb6d 32520	Version of 2sb6 2432 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)    &   ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)    &   ⊢ (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    ⇒   ⊢ (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜓)))
Theorem	wl-sbcom2d-lem1 32521*	Lemma used to prove wl-sbcom2d 32523. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
Theorem	wl-sbcom2d-lem2 32522*	Lemma used to prove wl-sbcom2d 32523. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑)))
Theorem	wl-sbcom2d 32523	Version of sbcom2 2433 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)    &   ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    &   ⊢ (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)    ⇒   ⊢ (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))
Theorem	wl-sbalnae 32524	A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Theorem	wl-sbal1 32525*	A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 32524 now. See also sbal1 2448. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Theorem	wl-sbal2 32526*	Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 32524 now. See also sbal2 2449. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
Theorem	wl-lem-exsb 32527*	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 32528	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 32529*	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 32530	This theorem extends alanimi 1734 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 32531	Version of mo2 2467 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv 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 32532	Closed form of mo2 2467 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 32533	Version of df-eu 2462 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate dv 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 32534	Closed form of df-eu 2462 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-euequ1f 32535	euequ1 2464 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
Theorem	wl-mo2t 32536*	Closed form of mo2 2467. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	wl-mo3t 32537*	Closed form of mo3 2495. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	wl-sb8eut 32538	Substitution of variable in universal quantifier. Closed form of sb8eu 2491. (Contributed by Wolf Lammen, 11-Aug-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
Theorem	wl-sb8mot 32539	Substitution of variable in universal quantifier. Closed form of sb8mo 2492. This theorem relates to wl-mo3t 32537, 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 2367 and sbco 2400. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 32537 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 32537 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))
Axiom	ax-wl-11v 32540*	Version of ax-11 2021 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2021. It will later be converted into a theorem directly based on ax-11 2021. (Contributed by Wolf Lammen, 28-Jun-2019.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	wl-ax11-lem1 32541	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 32542*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
Theorem	wl-ax11-lem3 32543*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)
Theorem	wl-ax11-lem4 32544*	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	wl-ax11-lem5 32545	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))
Theorem	wl-ax11-lem6 32546*	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 32547	Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
⊢ (∀𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))
Theorem	wl-ax11-lem8 32548*	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 32549	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 32550*	We now have prepared everything. The unwanted variable 𝑢 is just in one place left. pm2.61 182 can be used in conjunction with wl-ax11-lem9 32549 to eliminate the second antecedent. Missing is something along the lines of ax-6 1875, 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-sbcom3 32551	Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. Copy of ~? sbcom3OLD with a shortened proof. Keep this theorem for a while here because an external reference to it exists. (Contributed by Giovanni Mascellani, 8-Apr-2018.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑧 / 𝑦]𝜑)
21.18  Mathbox for Brendan Leahy
Theorem	rabiun 32552*	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 32553*	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 32554	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 32555	Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)
Theorem	unceq 32556	Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
Theorem	curf 32557	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 32558	Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
Theorem	curfv 32559	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 32560	Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹‘𝐴)‘𝐵))
Theorem	curunc 32561	Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
Theorem	unccur 32562	Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)
⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)
Theorem	phpreu 32563*	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 32564*	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 32565*	Lemma for fin2so 32566. (Contributed by Brendan Leahy, 29-Jun-2019.)
⊢ ((𝑅 Or 𝑥 ∧ (𝑦 ∈ 𝑥 ∧ 𝑧 ∈ 𝑥)) → (𝑦𝑅𝑧 → {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑦} [⊊] {𝑤 ∈ 𝑥 ∣ 𝑤𝑅𝑧}))
Theorem	fin2so 32566	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 32567	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 32568	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 32569	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 32570	Half-angle rule for cosine. (Contributed by Brendan Leahy, 4-Aug-2018.)
⊢ (𝐴 ∈ (-π[,]π) → (cos‘(𝐴 / 2)) = (√‘((1 + (cos‘𝐴)) / 2)))
Theorem	tan2h 32571	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	pigt3 32572	π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ 3 < π
Theorem	lindsdom 32573	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 32574	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 32575	One direction of matunitlindf 32577. (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 32576	One direction of matunitlindf 32577. (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 32577	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 32578*	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 32579*	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 32580*	Lemma for poimir 32612- 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 32581*	Lemma for poimir 32612- 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 32582*	Lemma for poimir 32612 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 32583*	Lemma for poimir 32612 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 32584*	Lemma for poimir 32612 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 32585*	Lemma for poimir 32612 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 32586*	Lemma for poimir 32612, similar to poimirlem6 32585, 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 32587*	Lemma for poimir 32612, establishing that away from the opposite vertex the walks in poimirlem9 32588 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 32588*	Lemma for poimir 32612, 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)})))))
Theorem	poimirlem10 32589*	Lemma for poimir 32612 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇)))
Theorem	poimirlem11 32590*	Lemma for poimir 32612 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑈) = 0)    &   ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st ‘𝑈)) “ (1...𝑀)))
Theorem	poimirlem12 32591*	Lemma for poimir 32612 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑈) = 𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1)))    ⇒   ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st ‘𝑈)) “ (1...𝑀)))
Theorem	poimirlem13 32592*	Lemma for poimir 32612- 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    ⇒   ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
Theorem	poimirlem14 32593*	Lemma for poimir 32612- 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    ⇒   ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
Theorem	poimirlem15 32594*	Lemma for poimir 32612, that the face in poimirlem22 32601 is a face. (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    ⇒   ⊢ (𝜑 → ⟨⟨(1st ‘(1st ‘𝑇)), ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))⟩, (2nd ‘𝑇)⟩ ∈ 𝑆)
Theorem	poimirlem16 32595*	Lemma for poimir 32612 establishing the vertices of the simplex of poimirlem17 32596. (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
Theorem	poimirlem17 32596*	Lemma for poimir 32612 establishing existence for poimirlem18 32597. (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem18 32597*	Lemma for poimir 32612 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem19 32598*	Lemma for poimir 32612 establishing the vertices of the simplex in poimirlem20 32599. (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
Theorem	poimirlem20 32599*	Lemma for poimir 32612 establishing existence for poimirlem21 32600. (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem21 32600*	Lemma for poimir 32612 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.)
⊢ (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 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    ⇒   ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)