Home Metamath Proof ExplorerTheorem List (p. 309 of 457) < 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 (1-28785) Hilbert Space Explorer (28786-30308) Users' Mathboxes (30309-45683)

Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrsp0 30801 The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.)
-∞ = (0.‘ℝ*𝑠)

Theoremxrsp1 30802 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
+∞ = (1.‘ℝ*𝑠)

Theoremressmulgnn 30803 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐴 ⊆ (Base‘𝐺)    &    = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐴) → (𝑁(.g𝐻)𝑋) = (𝑁 𝑋))

Theoremressmulgnn0 30804 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐴 ⊆ (Base‘𝐺)    &    = (.g𝐺)    &   𝐼 = (invg𝐺)    &   (0g𝐺) = (0g𝐻)       ((𝑁 ∈ ℕ0𝑋𝐴) → (𝑁(.g𝐻)𝑋) = (𝑁 𝑋))

20.3.8.7  The extended nonnegative real numbers commutative monoid

Theoremxrge0base 30805 The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(0[,]+∞) = (Base‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge00 30806 The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
0 = (0g‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge0plusg 30807 The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
+𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge0le 30808 The "less than or equal to" relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.)
≤ = (le‘(ℝ*𝑠s (0[,]+∞)))

Theoremxrge0mulgnn0 30809 The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ ℕ0𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵))

Theoremxrge0addass 30810 Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶)))

Theoremxrge0addgt0 30811 The sum of nonnegative and positive numbers is positive. See addgtge0 11151. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵))

Theoremxrge0adddir 30812 Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))

Theoremxrge0adddi 30813 Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ·e (𝐴 +𝑒 𝐵)) = ((𝐶 ·e 𝐴) +𝑒 (𝐶 ·e 𝐵)))

Theoremxrge0npcan 30814 Extended nonnegative real version of npcan 10918. (Contributed by Thierry Arnoux, 9-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)

Theoremfsumrp0cl 30815* Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → Σ𝑘𝐴 𝐵 ∈ (0[,)+∞))

20.3.9  Algebra

20.3.9.1  Monoids Homomorphisms

Theoremabliso 30816 The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)
((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

20.3.9.2  Finitely supported group sums - misc additions

Theoremgsumsubg 30817 The group sum in a subgroup is the same as the group sum. (Contributed by Thierry Arnoux, 28-May-2023.)
𝐻 = (𝐺s 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵 ∈ (SubGrp‘𝐺))       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsumsra 30818 The group sum in a subring algebra is the same as the ring's group sum. (Contributed by Thierry Arnoux, 28-May-2023.)
𝐴 = ((subringAlg ‘𝑅)‘𝐵)    &   (𝜑𝐹𝑈)    &   (𝜑𝑅𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐵 ⊆ (Base‘𝑅))       (𝜑 → (𝑅 Σg 𝐹) = (𝐴 Σg 𝐹))

Theoremgsummpt2co 30819* Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐸𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐸)    &   𝐹 = (𝑥𝐴𝐷)       (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)))))

Theoremgsummpt2d 30820* Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19145. (Contributed by Thierry Arnoux, 27-Apr-2020.)
𝑧𝐶    &   𝑦𝜑    &   𝐵 = (Base‘𝑊)    &   (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ CMnd)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)       (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))

Theoremlmodvslmhm 30821* Scalar multiplication in a left module by a fixed element is a group homomorphism. (Contributed by Thierry Arnoux, 12-Jun-2023.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑌𝑉) → (𝑥𝐾 ↦ (𝑥 · 𝑌)) ∈ (𝐹 GrpHom 𝑊))

Theoremgsumvsmul1 30822* Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc1 19412, since every ring is a left module over itself. (Contributed by Thierry Arnoux, 12-Jun-2023.)
𝐵 = (Base‘𝑅)    &   𝑆 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑅)    &   (𝜑𝑅 ∈ LMod)    &   (𝜑𝑆 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐾)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = ((𝑆 Σg (𝑘𝐴𝑋)) · 𝑌))

Theoremgsummptres 30823* Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥 ∈ (𝐴𝐷)) → 𝐶 = 0 )       (𝜑 → (𝐺 Σg (𝑥𝐴𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴𝐷) ↦ 𝐶)))

Theoremgsummptres2 30824* Extend a finite group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 22-Jun-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥 ∈ (𝐴𝑆)) → 𝑌 = 0 )    &   (𝜑𝑆 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝑌𝐵)    &   (𝜑𝑆𝐴)       (𝜑 → (𝐺 Σg (𝑥𝐴𝑌)) = (𝐺 Σg (𝑥𝑆𝑌)))

Theoremgsumzresunsn 30825 Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝑌 = (𝐹𝑋)    &   (𝜑𝐹:𝐶𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝐴)    &   (𝜑𝑋𝐶)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))))       (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹𝐴)) + 𝑌))

Theoremgsumpart 30826* Express a group sum as a double sum, grouping along a (possibly infinite) partition. (Contributed by Thierry Arnoux, 22-Jun-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝑋𝑊)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   (𝜑Disj 𝑥𝑋 𝐶)    &   (𝜑 𝑥𝑋 𝐶 = 𝐴)       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))

Theoremgsumhashmul 30827* Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(𝐹 “ {𝑥})) · 𝑥))))

Theoremxrge0tsmsd 30828* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑠))), ℝ*, < ))       (𝜑 → (𝐺 tsums 𝐹) = {𝑆})

Theoremxrge0tsmsbi 30829 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))       (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = (𝐺 tsums 𝐹)))

Theoremxrge0tsmseq 30830 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝐶 ∈ (𝐺 tsums 𝐹))       (𝜑𝐶 = (𝐺 tsums 𝐹))

20.3.9.3  Centralizers and centers - misc additions

Theoremcntzun 30831 The centralizer of a union is the intersection of the centralizers. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑋𝐵𝑌𝐵) → (𝑍‘(𝑋𝑌)) = ((𝑍𝑋) ∩ (𝑍𝑌)))

Theoremcntzsnid 30832 The centralizer of the identity element is the whole base set. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)    &    0 = (0g𝑀)       (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵)

Theoremcntrcrng 30833 The center of a ring is a commutative ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝑍 = (𝑅s (Cntr‘(mulGrp‘𝑅)))       (𝑅 ∈ Ring → 𝑍 ∈ CRing)

20.3.9.4  Totally ordered monoids and groups

Syntaxcomnd 30834 Extend class notation with the class of all right ordered monoids.
class oMnd

Syntaxcogrp 30835 Extend class notation with the class of all right ordered groups.
class oGrp

Definitiondf-omnd 30836* Define class of all right ordered monoids. An ordered monoid is a monoid with a total ordering compatible with its operation. It is possible to use this definition also for left ordered monoids, by applying it to (oppg𝑀). (Contributed by Thierry Arnoux, 13-Mar-2018.)
oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}

Definitiondf-ogrp 30837 Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 13-Mar-2018.)
oGrp = (Grp ∩ oMnd)

Theoremisomnd 30838* A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (le‘𝑀)       (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))

Theoremisogrp 30839 A (left-)ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Theoremogrpgrp 30840 A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
(𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Theoremomndmnd 30841 A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd → 𝑀 ∈ Mnd)

Theoremomndtos 30842 A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd → 𝑀 ∈ Toset)

Theoremomndadd 30843 In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))

Theoremomndaddr 30844 In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       (((oppg𝑀) ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑍 + 𝑋) (𝑍 + 𝑌))

Theoremomndadd2d 30845 In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑𝑀 ∈ CMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))

Theoremomndadd2rd 30846 In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑 → (oppg𝑀) ∈ oMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))

Theoremsubmomnd 30847 A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)

Theoremxrge0omnd 30848 The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
(ℝ*𝑠s (0[,]+∞)) ∈ oMnd

Theoremomndmul2 30849 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))

Theoremomndmul3 30850 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℕ0)    &   (𝜑𝑁𝑃)    &   (𝜑𝑋𝐵)    &   (𝜑0 𝑋)       (𝜑 → (𝑁 · 𝑋) (𝑃 · 𝑋))

Theoremomndmul 30851 In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 𝑌)       (𝜑 → (𝑁 · 𝑋) (𝑁 · 𝑌))

Theoremogrpinv0le 30852 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))

Theoremogrpsub 30853 In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))

Theoremogrpaddlt 30854 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))

Theoremogrpaddltbi 30855 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Theoremogrpaddltrd 30856 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → (oppg𝐺) ∈ oGrp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 < 𝑌)       (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌))

Theoremogrpaddltrbid 30857 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → (oppg𝐺) ∈ oGrp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌)))

Theoremogrpsublt 30858 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) < (𝑌 𝑍))

Theoremogrpinv0lt 30859 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))

Theoremogrpinvlt 30860 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &   𝐼 = (invg𝐺)       (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝐼𝑌) < (𝐼𝑋)))

Theoremgsumle 30861 A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐹r 𝐺)       (𝜑 → (𝑀 Σg 𝐹) (𝑀 Σg 𝐺))

20.3.9.5  The symmetric group

Theoremsymgfcoeu 30862* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))

Theoremsymgcom 30863 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋𝐸) = ( I ↾ 𝐸))    &   (𝜑 → (𝑌𝐹) = ( I ↾ 𝐹))    &   (𝜑 → (𝐸𝐹) = ∅)    &   (𝜑 → (𝐸𝐹) = 𝐴)       (𝜑 → (𝑋𝑌) = (𝑌𝑋))

Theoremsymgcom2 30864 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 17-Nov-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅)       (𝜑 → (𝑋𝑌) = (𝑌𝑋))

Theoremsymgcntz 30865* All elements of a (finite) set of permutations commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (Cntz‘𝑆)    &   (𝜑𝐴𝐵)    &   (𝜑Disj 𝑥𝐴 dom (𝑥 ∖ I ))       (𝜑𝐴 ⊆ (𝑍𝐴))

Theoremodpmco 30866 The composition of two odd permutations is even. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (pmEven‘𝐷)       ((𝐷 ∈ Fin ∧ 𝑋 ∈ (𝐵𝐴) ∧ 𝑌 ∈ (𝐵𝐴)) → (𝑋𝑌) ∈ 𝐴)

Theoremsymgsubg 30867 The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑌))

Theorempmtrprfv2 30868 In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)

Theorempmtrcnel 30869 Composing a permutation 𝐹 with a transposition which results in moving at least one less point. Here the set of points moved by a permutation 𝐹 is expressed as dom (𝐹 ∖ I ). (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))       (𝜑 → dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∖ {𝐼}))

Theorempmtrcnel2 30870 Variation on pmtrcnel 30869. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))       (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))

Theorempmtrcnelor 30871 Composing a permutation 𝐹 with a transposition which results in moving one or two less points. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))    &   𝐸 = dom (𝐹 ∖ I )    &   𝐴 = dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )       (𝜑 → (𝐴 = (𝐸 ∖ {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 ∖ {𝐼})))

20.3.9.6  Transpositions

Theorempmtridf1o 30872 Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑𝑇:𝐴1-1-onto𝐴)

Theorempmtridfv1 30873 Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑋) = 𝑌)

Theorempmtridfv2 30874 Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))       (𝜑 → (𝑇𝑌) = 𝑋)

20.3.9.7  Permutation Signs

Theorempsgnid 30875 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)

Theorempsgndmfi 30876 For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑆 = (pmSgn‘𝐷)    &   𝐺 = (Base‘(SymGrp‘𝐷))       (𝐷 ∈ Fin → 𝑆 Fn 𝐺)

Theorempmtrto1cl 30877 Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑇 = (pmTrsp‘𝐷)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇)

Theorempsgnfzto1stlem 30878* Lemma for psgnfzto1st 30883. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖𝐾, (𝑖 − 1), 𝑖)))))

Theoremfzto1stfv1 30879* Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼𝐷 → (𝑃‘1) = 𝐼)

Theoremfzto1st1 30880* Special case where the permutation defined in psgnfzto1st 30883 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷))

Theoremfzto1st 30881* The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷𝑃𝐵)

Theoremfzto1stinvn 30882* Value of the inverse of our permutation 𝑃 at 𝐼. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷 → (𝑃𝐼) = 1)

Theorempsgnfzto1st 30883* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑆 = (pmSgn‘𝐷)       (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))

20.3.9.8  Permutation cycles

Syntaxctocyc 30884 Extend class notation with the permutation cycle builder.
class toCyc

Definitiondf-tocyc 30885* Define a convenience permutation cycle builder. Given a list of elements to be cycled, in the form of a word, this function produces the corresponding permutation cycle. See definition in [Lang] p. 30. (Contributed by Thierry Arnoux, 19-Sep-2023.)
toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑𝑢:dom 𝑢1-1𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ 𝑤))))

Theoremtocycval 30886* Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)       (𝐷𝑉𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷𝑢:dom 𝑢1-1𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ 𝑤))))

Theoremtocycfv 30887 Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → (𝐶𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊)))

Theoremtocycfvres1 30888 A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → ((𝐶𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ 𝑊))

Theoremtocycfvres2 30889 A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → ((𝐶𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊)))

Theoremcycpmfvlem 30890 Lemma for cycpmfv1 30891 and cycpmfv2 30892. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑁 ∈ (0..^(♯‘𝑊)))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (((𝑊 cyclShift 1) ∘ 𝑊)‘(𝑊𝑁)))

Theoremcycpmfv1 30891 Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑁 ∈ (0..^((♯‘𝑊) − 1)))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (𝑊‘(𝑁 + 1)))

Theoremcycpmfv2 30892 Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 0 < (♯‘𝑊))    &   (𝜑𝑁 = ((♯‘𝑊) − 1))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (𝑊‘0))

Theoremcycpmfv3 30893 Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑋𝐷)    &   (𝜑 → ¬ 𝑋 ∈ ran 𝑊)       (𝜑 → ((𝐶𝑊)‘𝑋) = 𝑋)

Theoremcycpmcl 30894 Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → (𝐶𝑊) ∈ (Base‘𝑆))

Theoremtocycf 30895* The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)       (𝐷𝑉𝐶:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶𝐵)

Theoremtocyc01 30896 Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.)
𝐶 = (toCyc‘𝐷)       ((𝐷𝑉𝑊 ∈ (dom 𝐶 ∩ (♯ “ {0, 1}))) → (𝐶𝑊) = ( I ↾ 𝐷))

Theoremcycpm2tr 30897 A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑇 = (pmTrsp‘𝐷)       (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) = (𝑇‘{𝐼, 𝐽}))

Theoremcycpm2cl 30898 Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))

Theoremcyc2fv1 30899 Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)

Theoremcyc2fv2 30900 Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)

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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45683
 Copyright terms: Public domain < Previous  Next >