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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-toplnd 26701* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopLnd

16.16  Mathbox for Steve Rodriguez

16.16.1  Miscellanea

Theoremiso0 26702 The empty set is an isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)

Theoremssrecnpr 26703 is a subset of both and . (Contributed by Steve Rodriguez, 22-Nov-2015.)

Theoremseff 26704 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 26705 The infinity ball in the absolute value metric is just the whole space. analog of blpnf 17786. (Contributed by Steve Rodriguez, 8-Nov-2015.)

16.16.2  Function operations

Theoremcaofcan 26706* Transfer a cancellation law like mulcan 9285 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)

Theoremofsubid 26707 Function analog of subid 8947. (Contributed by Steve Rodriguez, 5-Nov-2015.)

Theoremofmul12 26708 Function analog of mul12 8858. (Contributed by Steve Rodriguez, 13-Nov-2015.)

Theoremofdivrec 26709 Function analog of divrec 9320, a division analog of ofnegsub 9624. (Contributed by Steve Rodriguez, 3-Nov-2015.)

Theoremofdivcan4 26710 Function analog of divcan4 9329. (Contributed by Steve Rodriguez, 4-Nov-2015.)

Theoremofdivdiv2 26711 Function analog of divdiv2 9352. (Contributed by Steve Rodriguez, 23-Nov-2015.)

16.16.3  Calculus

Theoremlhe4.4ex1a 26712 Example of the Fundamental Theorem of Calculus, part two (ftc2 19223): . Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 19223 as simply the "Fundamental Theorem of Calculus", then ftc1 19221 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)

Theoremdvsconst 26713 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 26714 Derivative of the identity function on the reals or complexes. (Contributed by Steve Rodriguez, 11-Nov-2015.)

Theoremdvsef 26715 Derivative of the exponential function on the reals or complexes. (Contributed by Steve Rodriguez, 12-Nov-2015.)

Theoremexpgrowthi 26716* Exponential growth and decay model. See expgrowth 26718 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)

Theoremdvconstbi 26717* The derivative of a function on is zero iff it is a constant function. Roughly a biconditional analog of dvconst 19098 and dveq0 19179. Corresponds to integration formula " " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)

Theoremexpgrowth 26718* 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 26716 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 26716 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

16.17  Mathbox for Andrew Salmon

16.17.1  Principia Mathematica * 10

Theorempm10.12 26719* 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 26720 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.251 26721 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.252 26722 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.253 26723 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theoremalbitr 26724 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.42 26725 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm10.52 26726* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.53 26727 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.541 26728* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.542 26729* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.55 26730 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.56 26731 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm10.57 26732 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

16.17.2  Principia Mathematica * 11

Theorem2alanimi 26733 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2al2imi 26734 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremstdpc4-2 26735 Theorem *11.1 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.11 26736 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorempm11.12 26737* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)

Theorem2exnaln 26738 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2nexaln 26739 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.21vv 26740* 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 26741 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 26742 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 26743 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 26744 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorema4sbce-2 26745 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.33-2 26746 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 26747* 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 26748* 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 26749* 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 26750* 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 26751 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem2exanali 26752 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremaaanv 26753* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 1811. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.57 26754* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.58 26755* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.59 26756* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Theorempm11.6 26757* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Theorempm11.61 26758* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.62 26759* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.63 26760 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.7 26761 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Theorempm11.71 26762* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

16.17.3  Predicate Calculus

Theoremsbeqal1 26763* If always implies , then is true. (Contributed by Andrew Salmon, 2-Jun-2011.)

Theoremsbeqal1i 26764* Suppose you know implies , assuming and are distinct. Then, . (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremsbeqal2i 26765* If implies , then we can infer . (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremsbeqalbi 26766* When both and and and are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)

Theoremax4567 26767 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1536 as the inference rule. This proof extends the idea of ax467 1752 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)

Theoremax4567to4 26768 Re-derivation of ax-4 1692 from ax4567 26767. Note that ax-9 1684 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax4567to5 26769 Re-derivation of ax-5o 1694 from ax4567 26767. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax4567to6 26770 Re-derivation of ax-6o 1697 from ax4567 26767. Note that neither ax-6o 1697 nor ax-7 1535 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax4567to7 26771 Re-derivation of ax-7 1535 from ax4567 26767. Note that ax-7 1535 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)

Theoremax10ext 26772* This theorem shows that, given axext4 2237, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.)

Theoremax10-16 26773* This theorem shows that, given ax-16 1926, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.)

16.17.4  Principia Mathematica * 13 and * 14

Theorempm13.13a 26774 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 26775 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.14 26776 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.192 26777* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Theorempm13.193 26778 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.194 26779 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorempm13.195 26780* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 2945. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Theorempm13.196a 26781* 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 26782* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theorem2sbc5g 26783* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremiotain 26784 Equivalence between two different forms of . (Contributed by Andrew Salmon, 15-Jul-2011.)

Theoremiotaexeu 26785 The iota class exists. This theorem does not require ax-nul 4046 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotasbc 26786* 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 26787* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theorempm14.12 26788* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theorempm14.122a 26789* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Theorempm14.122b 26790* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Theorempm14.122c 26791* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Theorempm14.123a 26792* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Theorempm14.123b 26793* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Theorempm14.123c 26794* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Theorempm14.18 26795 Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotaequ 26796* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotavalb 26797* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6154. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremiotasbc5 26798* Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theorempm14.24 26799* Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Theoremiotavalsb 26800* Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

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