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Type | Label | Description |
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Statement | ||
Theorem | extoimad 36601* | If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | imo72b2lem0 36602* | Lemma for imo72b2 36613. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | suprleubrd 36603* | Natural deduction form of specialized suprleub 10570. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | imo72b2lem2 36604* | Lemma for imo72b2 36613. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | syldbl2 36605 | Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | funfvima2d 36606 | A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | suprlubrd 36607* | Natural deduction form of specialized suprlub 10568. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | imo72b2lem1 36608* | Lemma for imo72b2 36613. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | lemuldiv3d 36609 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | lemuldiv4d 36610 | 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | hypstkd 36611 | Natural deductionm, stacks an hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | rspcdvinvd 36612* | If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | imo72b2 36613* | IMO 1972 B2. (14th International Mathemahics Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.) |
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This section formalizes theorems necessary to reproduce the equality and inequality generator described in "Neural Theorem Proving on Inequality Problems" http://aitp-conference.org/2020/abstract/paper_18.pdf. Other theorems required: 0red 9641 1red 9655 readdcld 9667 remulcld 9668 eqcomd 2456. | ||
Theorem | int-addcomd 36614 | AdditionCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-addassocd 36615 | AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-addsimpd 36616 | AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-mulcomd 36617 | MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-mulassocd 36618 | MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-mulsimpd 36619 | MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-leftdistd 36620 | AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-rightdistd 36621 | AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-sqdefd 36622 | SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-mul11d 36623 | First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-mul12d 36624 | Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-add01d 36625 | First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-add02d 36626 | Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-sqgeq0d 36627 | SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-eqprincd 36628 | PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-eqtransd 36629 | EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-eqmvtd 36630 | EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-eqineqd 36631 | EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-ineqmvtd 36632 | IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-ineq1stprincd 36633 | FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-ineq2ndprincd 36634 | SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | int-ineqtransd 36635 | InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 11062 addcomli 9822 00id 9805 addid1i 9817 addid2i 9818 eqid 2450 dec0h 11064 decadd 11089 decaddc 11090. | ||
Theorem | unitadd 36636 | Theorem used in conjunction with decaddc 11090 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | 5p5e10b 36637 | 5 + 5 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | 6p4e10b 36638 | 6 + 4 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | 7p3e10b 36639 | 7 + 3 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | 8p2e10b 36640 | 8 + 2 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | 9p1e10b 36641 | 9 + 1 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.) |
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Theorem | gsumws3 36642 | Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.) |
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Theorem | gsumws4 36643 | Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.) |
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Theorem | amgm2d 36644 |
Arithmetic-geometric mean inequality for ![]() ![]() ![]() |
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Theorem | amgm3d 36645 |
Arithmetic-geometric mean inequality for ![]() ![]() ![]() |
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Theorem | amgm4d 36646 |
Arithmetic-geometric mean inequality for ![]() ![]() ![]() |
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Theorem | nanorxor 36647 | 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.) |
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Theorem | undisjrab 36648 | Union of two disjoint restricted class abstractions; compare unrab 3713. (Contributed by Steve Rodriguez, 28-Feb-2020.) |
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Theorem | iso0 36649 |
The empty set is an ![]() ![]() ![]() |
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Theorem | ssrecnpr 36650 |
![]() ![]() ![]() |
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Theorem | seff 36651 |
Let set ![]() ![]() ![]() ![]() |
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Theorem | sblpnf 36652 |
The infinity ball in the absolute value metric is just the whole space.
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Theorem | prmunb2 36653* |
The primes are unbounded. This generalizes prmunb 14851 to real ![]() |
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Theorem | isprm7 36654* |
One need only check prime divisors of ![]() ![]() ![]() ![]() |
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Theorem | dvgrat 36655* |
Ratio test for divergence of a complex infinite series. See e.g. remark
"if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cvgdvgrat 36656* |
Ratio test for convergence and divergence of a complex infinite series.
If the ratio ![]() ![]() ![]() ![]() (It also demonstrates how to use climi2 13568 and absltd 13484 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191, and how to use r19.29a 2931 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 2878 at https://groups.google.com/forum/#!topic/metamath/2RPikOiXLMo.) (Contributed by Steve Rodriguez, 28-Feb-2020.) |
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Theorem | radcnvrat 36657* |
Let ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | reldvds 36658 | The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | nznngen 36659 | All positive integers in the set of multiples of n, nℤ, are the absolute value of n or greater. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | nzss 36660 | The set of multiples of m, mℤ, is a subset of those of n, nℤ, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | nzin 36661 | The intersection of the set of multiples of m, mℤ, and those of n, nℤ, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | nzprmdif 36662 | Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | hashnzfz 36663 | Special case of hashdvds 14716: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | hashnzfz2 36664 | Special case of hashnzfz 36663: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | hashnzfzclim 36665* |
As the upper bound ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | caofcan 36666* | Transfer a cancellation law like mulcan 10246 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.) |
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Theorem | ofsubid 36667 | Function analogue of subid 9890. (Contributed by Steve Rodriguez, 5-Nov-2015.) |
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Theorem | ofmul12 36668 | Function analogue of mul12 9796. (Contributed by Steve Rodriguez, 13-Nov-2015.) |
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Theorem | ofdivrec 36669 | Function analogue of divrec 10283, a division analogue of ofnegsub 10604. (Contributed by Steve Rodriguez, 3-Nov-2015.) |
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Theorem | ofdivcan4 36670 | Function analogue of divcan4 10292. (Contributed by Steve Rodriguez, 4-Nov-2015.) |
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Theorem | ofdivdiv2 36671 | Function analogue of divdiv2 10316. (Contributed by Steve Rodriguez, 23-Nov-2015.) |
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Theorem | lhe4.4ex1a 36672 |
Example of the Fundamental Theorem of Calculus, part two (ftc2 22989):
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvsconst 36673 |
Derivative of a constant function on the real or complex numbers. The
function may return a complex ![]() ![]() ![]() |
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Theorem | dvsid 36674 | Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.) |
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Theorem | dvsef 36675 | Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.) |
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Theorem | expgrowthi 36676* | Exponential growth and decay model. See expgrowth 36678 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.) |
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Theorem | dvconstbi 36677* |
The derivative of a function on ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | expgrowth 36678* |
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 36676 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 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 36676 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.) |
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Syntax | cbcc 36679 | Extend class notation to include the generalized binomial coefficient operation. |
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Definition | df-bcc 36680* | Define a generalized binomial coefficient operation, which unlike df-bc 12485 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
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Theorem | bccval 36681 |
Value of the generalized binomial coefficient, ![]() ![]() |
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Theorem | bcccl 36682 | Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
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Theorem | bcc0 36683 |
The generalized binomial coefficient ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bccp1k 36684 |
Generalized binomial coefficient: ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bccm1k 36685 |
Generalized binomial coefficient: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | bccn0 36686 |
Generalized binomial coefficient: ![]() ![]() |
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Theorem | bccn1 36687 |
Generalized binomial coefficient: ![]() ![]() |
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Theorem | bccbc 36688 | The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
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Theorem | uzmptshftfval 36689* |
When ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvradcnv2 36690* |
The radius of convergence of the (formal) derivative ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | binomcxplemwb 36691 | Lemma for binomcxp 36700. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
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Theorem | binomcxplemnn0 36692* |
Lemma for binomcxp 36700. When ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | binomcxplemrat 36693* |
Lemma for binomcxp 36700. As ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | binomcxplemfrat 36694* |
Lemma for binomcxp 36700. binomcxplemrat 36693 implies that when ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | binomcxplemradcnv 36695* |
Lemma for binomcxp 36700. By binomcxplemfrat 36694 and radcnvrat 36657 the
radius of convergence of power series
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | binomcxplemdvbinom 36696* |
Lemma for binomcxp 36700. By the power and chain rules, calculate
the
derivative of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | binomcxplemcvg 36697* |
Lemma for binomcxp 36700. The sum in binomcxplemnn0 36692 and its
derivative (see the next theorem, binomcxplemdvsum 36698) converge, as
long as their base ![]() |
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Theorem | binomcxplemdvsum 36698* | Lemma for binomcxp 36700. The derivative of the generalized sum in binomcxplemnn0 36692. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
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Theorem | binomcxplemnotnn0 36699* |
Lemma for binomcxp 36700. When ![]() ![]() ![]()
pserdv2 23378 gives the derivative of
Finally, let |
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Theorem | binomcxp 36700* |
Generalize the binomial theorem binom 13881 to positive real summand
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