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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-cytp 27501* The Nth cyclotomic polynomial is the polynomial which has as its zeros precisely the primitive Nth roots of unity. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP mulGrpPoly1fld g mulGrpflds var1fldPoly1fldalgScPoly1fld

Theoremisdomn3 27502 Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
mulGrp       Domn SubMnd

Theoremmon1pid 27503 Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Poly1              Monic1p       deg1        NzRing

Theoremmon1psubm 27504 Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Poly1       Monic1p       mulGrp       NzRing SubMnd

Theoremdeg1mhm 27505 Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
deg1               Poly1              mulGrps        flds        Domn MndHom

Theoremcytpfn 27506 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP

Theoremcytpval 27507* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
mulGrpflds               Poly1fld       var1fld       mulGrp              algSc       CytP g

19.16.63  Miscellaneous topology

Theoremfgraphopab 27508* Express a function as a subset of the cross product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremfgraphxp 27509* Express a function as a subset of the cross product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremhausgraph 27510 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Syntaxctopsep 27511 The class of separable toplogies.
TopSep

Syntaxctoplnd 27512 The class of Lindelöf toplogies.
TopLnd

Definitiondf-topsep 27513* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopSep

Definitiondf-toplnd 27514* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopLnd

19.17  Mathbox for Steve Rodriguez

19.17.1  Miscellanea

Theoremiso0 27515 The empty set is an isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)

Theoremssrecnpr 27516 is a subset of both and . (Contributed by Steve Rodriguez, 22-Nov-2015.)

Theoremseff 27517 Let set be the reals or complexes. Then the exponential function restricted to is a mapping from to . (Contributed by Steve Rodriguez, 6-Nov-2015.)

Theoremsblpnf 27518 The infinity ball in the absolute value metric is just the whole space. analog of blpnf 18429. (Contributed by Steve Rodriguez, 8-Nov-2015.)

19.17.2  Function operations

Theoremcaofcan 27519* Transfer a cancellation law like mulcan 9661 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)

Theoremofsubid 27520 Function analog of subid 9323. (Contributed by Steve Rodriguez, 5-Nov-2015.)

Theoremofmul12 27521 Function analog of mul12 9234. (Contributed by Steve Rodriguez, 13-Nov-2015.)

Theoremofdivrec 27522 Function analog of divrec 9696, a division analog of ofnegsub 10000. (Contributed by Steve Rodriguez, 3-Nov-2015.)

Theoremofdivcan4 27523 Function analog of divcan4 9705. (Contributed by Steve Rodriguez, 4-Nov-2015.)

Theoremofdivdiv2 27524 Function analog of divdiv2 9728. (Contributed by Steve Rodriguez, 23-Nov-2015.)

19.17.3  Calculus

Theoremlhe4.4ex1a 27525 Example of the Fundamental Theorem of Calculus, part two (ftc2 19930): . Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 19930 as simply the "Fundamental Theorem of Calculus", then ftc1 19928 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)

Theoremdvsconst 27526 Derivative of a constant function on the reals or complexes. The function may return a complex even if is . (Contributed by Steve Rodriguez, 11-Nov-2015.)

Theoremdvsid 27527 Derivative of the identity function on the reals or complexes. (Contributed by Steve Rodriguez, 11-Nov-2015.)

Theoremdvsef 27528 Derivative of the exponential function on the reals or complexes. (Contributed by Steve Rodriguez, 12-Nov-2015.)

Theoremexpgrowthi 27529* Exponential growth and decay model. See expgrowth 27531 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)

Theoremdvconstbi 27530* The derivative of a function on is zero iff it is a constant function. Roughly a biconditional analog of dvconst 19805 and dveq0 19886. Corresponds to integration formula " " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)

Theoremexpgrowth 27531* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 27529 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as , C as , and ky as . is the constant function that maps any real or complex input to k and is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 27529 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

19.18  Mathbox for Andrew Salmon

19.18.1  Principia Mathematica * 10

Theorempm10.12 27532* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.14 27533 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.251 27534 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.252 27535 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.253 27536 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theoremalbitr 27537 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.42 27538 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.52 27539* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.53 27540 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.541 27541* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.542 27542* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.55 27543 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.56 27544 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.57 27545 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

19.18.2  Principia Mathematica * 11

Theorem2alanimi 27546 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2al2imi 27547 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremstdpc4-2 27548 Theorem *11.1 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.11 27549 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm11.12 27550* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorem2exnaln 27551 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2nexaln 27552 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.21vv 27553* Compare theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2alim 27554 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2albi 27555 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2exim 27556 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2exbi 27557 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremspsbce-2 27558 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.33-2 27559 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.36vv 27560* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)

Theorem19.31vv 27561* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.37vv 27562* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.28vv 27563* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.52 27564 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2exanali 27565 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremaaanv 27566* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 1907. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.57 27567* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.58 27568* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.59 27569* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Theorempm11.6 27570* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Theorempm11.61 27571* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.62 27572* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.63 27573 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.7 27574 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.71 27575* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

19.18.3  Predicate Calculus

Theoremsbeqal1 27576* If always implies , then is true. (Contributed by Andrew Salmon, 2-Jun-2011.)

Theoremsbeqal1i 27577* Suppose you know implies , assuming and are distinct. Then, . (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremsbeqal2i 27578* If implies , then we can infer . (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremsbeqalbi 27579* When both and and and are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)

Theoremax4567 27580 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1556 as the inference rule. This proof extends the idea of ax467 2248 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)

Theoremax4567to4 27581 Re-derivation of sp 1764 from ax4567 27580. Note that ax9 1954 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax4567to5 27582 Re-derivation of ax5o 1766 from ax4567 27580. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax4567to6 27583 Re-derivation of ax6o 1767 from ax4567 27580. Note that neither ax6o 1767 nor ax-7 1750 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax4567to7 27584 Re-derivation of ax-7 1750 from ax4567 27580. Note that ax-7 1750 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax10ext 27585* This theorem shows that, given axext4 2422, we can derive a version of ax10 2026. However, it is weaker than ax10 2026 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.)

19.18.4  Principia Mathematica * 13 and * 14

Theorempm13.13a 27586 One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.13b 27587 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.14 27588 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.192 27589* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Theorempm13.193 27590 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.194 27591 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.195 27592* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3187. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Theorempm13.196a 27593* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorem2sbc6g 27594* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorem2sbc5g 27595* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremiotain 27596 Equivalence between two different forms of . (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremiotaexeu 27597 The iota class exists. This theorem does not require ax-nul 4340 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotasbc 27598* Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define in terms of a function of . Their definition differs in that a function of evaluates to "false" when there isn't a single that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotasbc2 27599* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theorempm14.12 27600* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)

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