Home Metamath Proof ExplorerTheorem List (p. 326 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

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

Theoremwl-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)))

Theoremwl-nfeqfb 32502 Extend nfeqf 2289 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
(Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))

Theoremwl-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)

Theoremwl-equsald 32504 Deduction version of equsal 2279. (Contributed by Wolf Lammen, 27-Jul-2019.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Theoremwl-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.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremwl-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 = 𝑦𝜑) ↔ 𝜑))

Theoremwl-equsalcom 32507 This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑦 = 𝑥𝜑))

Theoremwl-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 = 𝑦𝜑)       𝜑

Theoremwl-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)))

Theoremwl-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](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

Theoremwl-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](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))

Theoremwl-sb8t 32512 Substitution of variable in universal quantifier. Closed form of sb8 2412. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sb8et 32513 Substitution of variable in universal quantifier. Closed form of sb8e 2413. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-sbhbt 32514 Closed form of sbhb 2426. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Theoremwl-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.)
(∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Theoremwl-equsb3 32516 equsb3 2420 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
(¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))

Theoremwl-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 = 𝑧𝑦 = 𝑧))

Theoremwl-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 = 𝑦𝜑)))

Theoremwl-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 = 𝑦𝜑)))

Theoremwl-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 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))

Theoremwl-sbcom2d-lem1 32521* Lemma used to prove wl-sbcom2d 32523. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))

Theoremwl-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 ↔ ∀𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) → 𝜑)))

Theoremwl-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))

Theoremwl-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))

Theoremwl-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))

Theoremwl-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))

Theoremwl-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 = 𝑦𝜑)))

Theoremwl-lem-nexmo 32528 This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))

Theoremwl-lem-moexsb 32529* The antecedent 𝑥(𝜑𝑥 = 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(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremwl-alanbii 32530 This theorem extends alanimi 1734 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
(𝜑 ↔ (𝜓𝜒))       (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))

Theoremwl-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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Theoremwl-mo2tf 32532 Closed form of mo2 2467 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))

Theoremwl-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 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-euequ1f 32535 euequ1 2464 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Theoremwl-mo2t 32536* Closed form of mo2 2467. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))

Theoremwl-mo3t 32537* Closed form of mo3 2495. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Theoremwl-sb8eut 32538 Substitution of variable in universal quantifier. Closed form of sb8eu 2491. (Contributed by Wolf Lammen, 11-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))

Theoremwl-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.)

(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))

Axiomax-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.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremwl-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))

Theoremwl-ax11-lem2 32542* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)

Theoremwl-ax11-lem3 32543* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)

Theoremwl-ax11-lem4 32544* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-ax11-lem5 32545 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Theoremwl-ax11-lem6 32546* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑥𝑦𝜑))

Theoremwl-ax11-lem7 32547 Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥(¬ ∀𝑥 𝑥 = 𝑦𝜑) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜑))

Theoremwl-ax11-lem8 32548* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑢𝑥[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝑥𝜑))

Theoremwl-ax11-lem9 32549 The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝑥𝜑 ↔ ∀𝑥𝑦𝜑))

Theoremwl-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 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)))

Theoremwl-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

Theoremrabiun 32552* Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
{𝑥 𝑦𝐴 𝐵𝜑} = 𝑦𝐴 {𝑥𝐵𝜑}

Theoremiundif1 32553* Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)

Theoremimadifss 32554 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 32555 Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(𝐴 = 𝐵 → curry 𝐴 = curry 𝐵)

Theoremunceq 32556 Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Theoremcurf 32557 Functional property of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶𝑚 𝐵))

Theoremuncf 32558 Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(𝐹:𝐴⟶(𝐶𝑚 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Theoremcurfv 32559 Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
(((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))

Theoremuncov 32560 Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴uncurry 𝐹𝐵) = ((𝐹𝐴)‘𝐵))

Theoremcurunc 32561 Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)

Theoremunccur 32562 Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)
((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = 𝐹)

Theoremphpreu 32563* Theorem related to pigeonhole principle. (Contributed by Brendan Leahy, 21-Aug-2020.)
((𝐴 ∈ Fin ∧ 𝐴𝐵) → (∀𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥 = 𝐶))

Theoremfinixpnum 32564* 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 32565* Lemma for fin2so 32566. (Contributed by Brendan Leahy, 29-Jun-2019.)
((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑧𝑥)) → (𝑦𝑅𝑧 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑧}))

Theoremfin2so 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 ∈ FinII𝑅 Or 𝐴) → 𝐴 ∈ Fin)

Theoremltflcei 32567 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 32568 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 32569 Half-angle rule for sine. (Contributed by Brendan Leahy, 3-Aug-2018.)
(𝐴 ∈ (0[,](2 · π)) → (sin‘(𝐴 / 2)) = (√‘((1 − (cos‘𝐴)) / 2)))

Theoremcos2h 32570 Half-angle rule for cosine. (Contributed by Brendan Leahy, 4-Aug-2018.)
(𝐴 ∈ (-π[,]π) → (cos‘(𝐴 / 2)) = (√‘((1 + (cos‘𝐴)) / 2)))

Theoremtan2h 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)))))

Theorempigt3 32572 π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)
3 < π

Theoremlindsdom 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))) → 𝑋𝐼)

Theoremlindsenlbs 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))) ∧ 𝑋𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))

Theoremmatunitlindflem1 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) = (0g𝑅)))

Theoremmatunitlindflem2 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))

Theoremmatunitlindf 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)))

Theoremptrest 32578* 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 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))       ((𝑆𝑅𝑃𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)

Theorempoimirlem1 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.)
(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 32581* Lemma for poimir 32612- 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 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.)
(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 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.)
(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 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.)
(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 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.)
(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 32586* Lemma for poimir 32612, similar to poimirlem6 32585, 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 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.)
(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 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.)
(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 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.)
(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 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.)
(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 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.)
(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 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.)
(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 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.)
(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 32594* Lemma for poimir 32612, that the face in poimirlem22 32601 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 32595* Lemma for poimir 32612 establishing the vertices of the simplex of poimirlem17 32596. (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 32596* Lemma for poimir 32612 establishing existence for poimirlem18 32597. (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 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.)
(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 32598* Lemma for poimir 32612 establishing the vertices of the simplex in poimirlem20 32599. (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 32599* Lemma for poimir 32612 establishing existence for poimirlem21 32600. (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 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.)
(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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 42301-42360
 Copyright terms: Public domain < Previous  Next >