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Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnzprmdif 40101 Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(𝜑𝑀 ∈ ℙ)    &   (𝜑𝑁 ∈ ℙ)    &   (𝜑𝑀𝑁)       (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)})))

Theoremhashnzfz 40102 Special case of hashdvds 15967: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝐾 ∈ (ℤ‘(𝐽 − 1)))       (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁))))

Theoremhashnzfz2 40103 Special case of hashnzfz 40102: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ ℕ)       (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁)))

Theoremhashnzfzclim 40104* As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 40102 increases, the resulting count of multiples tends to (𝐾 / 𝑀) —that is, there are approximately (𝐾 / 𝑀) multiples of 𝑀 in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐽 ∈ ℤ)       (𝜑 → (𝑘 ∈ (ℤ‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) ⇝ (1 / 𝑀))

20.32.4  Function operations

Theoremcaofcan 40105* Transfer a cancellation law like mulcan 11077 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑇)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))       (𝜑 → ((𝐹𝑓 𝑅𝐺) = (𝐹𝑓 𝑅𝐻) ↔ 𝐺 = 𝐻))

Theoremofsubid 40106 Function analogue of subid 10705. (Contributed by Steve Rodriguez, 5-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ) → (𝐹𝑓𝐹) = (𝐴 × {0}))

Theoremofmul12 40107 Function analogue of mul12 10604. (Contributed by Steve Rodriguez, 13-Nov-2015.)
(((𝐴𝑉𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶ℂ ∧ 𝐻:𝐴⟶ℂ)) → (𝐹𝑓 · (𝐺𝑓 · 𝐻)) = (𝐺𝑓 · (𝐹𝑓 · 𝐻)))

Theoremofdivrec 40108 Function analogue of divrec 11114, a division analogue of ofnegsub 11436. (Contributed by Steve Rodriguez, 3-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → (𝐹𝑓 · ((𝐴 × {1}) ∘𝑓 / 𝐺)) = (𝐹𝑓 / 𝐺))

Theoremofdivcan4 40109 Function analogue of divcan4 11125. (Contributed by Steve Rodriguez, 4-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → ((𝐹𝑓 · 𝐺) ∘𝑓 / 𝐺) = 𝐹)

Theoremofdivdiv2 40110 Function analogue of divdiv2 11152. (Contributed by Steve Rodriguez, 23-Nov-2015.)
(((𝐴𝑉𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶(ℂ ∖ {0}) ∧ 𝐻:𝐴⟶(ℂ ∖ {0}))) → (𝐹𝑓 / (𝐺𝑓 / 𝐻)) = ((𝐹𝑓 · 𝐻) ∘𝑓 / 𝐺))

20.32.5  Calculus

Theoremlhe4.4ex1a 40111 Example of the Fundamental Theorem of Calculus, part two (ftc2 24360): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 24360 as simply the "Fundamental Theorem of Calculus", then ftc1 24358 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3)

Theoremdvsconst 40112 Derivative of a constant function on the real or complex numbers. The function may return a complex 𝐴 even if 𝑆 is . (Contributed by Steve Rodriguez, 11-Nov-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐴 ∈ ℂ) → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0}))

Theoremdvsid 40113 Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.)
(𝑆 ∈ {ℝ, ℂ} → (𝑆 D ( I ↾ 𝑆)) = (𝑆 × {1}))

Theoremdvsef 40114 Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.)
(𝑆 ∈ {ℝ, ℂ} → (𝑆 D (exp ↾ 𝑆)) = (exp ↾ 𝑆))

Theoremexpgrowthi 40115* Exponential growth and decay model. See expgrowth 40117 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐾 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   𝑌 = (𝑡𝑆 ↦ (𝐶 · (exp‘(𝐾 · 𝑡))))       (𝜑 → (𝑆 D 𝑌) = ((𝑆 × {𝐾}) ∘𝑓 · 𝑌))

Theoremdvconstbi 40116* The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 24233 and dveq0 24316. Corresponds to integration formula "∫0 d𝑥 = 𝐶 " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑌:𝑆⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝑌) = 𝑆)       (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑆 × {𝑐})))

Theoremexpgrowth 40117* 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 40115 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 (𝑆 D 𝑌), C as 𝑐, and ky as ((𝑆 × {𝐾}) ∘𝑓 · 𝑌). (𝑆 × {𝐾}) is the constant function that maps any real or complex input to k and 𝑓 · 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 40115 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐾 ∈ ℂ)    &   (𝜑𝑌:𝑆⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝑌) = 𝑆)       (𝜑 → ((𝑆 D 𝑌) = ((𝑆 × {𝐾}) ∘𝑓 · 𝑌) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑡𝑆 ↦ (𝑐 · (exp‘(𝐾 · 𝑡))))))

20.32.6  The generalized binomial coefficient operation

Syntaxcbcc 40118 Extend class notation to include the generalized binomial coefficient operation.
class C𝑐

Definitiondf-bcc 40119* Define a generalized binomial coefficient operation, which unlike df-bc 13477 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))

Theorembccval 40120 Value of the generalized binomial coefficient, 𝐶 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))

Theorembcccl 40121 Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ)

Theorembcc0 40122 The generalized binomial coefficient 𝐶 choose 𝐾 is zero iff 𝐶 is an integer between zero and (𝐾 − 1) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → ((𝐶C𝑐𝐾) = 0 ↔ 𝐶 ∈ (0...(𝐾 − 1))))

Theorembccp1k 40123 Generalized binomial coefficient: 𝐶 choose (𝐾 + 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → (𝐶C𝑐(𝐾 + 1)) = ((𝐶C𝑐𝐾) · ((𝐶𝐾) / (𝐾 + 1))))

Theorembccm1k 40124 Generalized binomial coefficient: 𝐶 choose (𝐾 − 1), when 𝐶 is not (𝐾 − 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}))    &   (𝜑𝐾 ∈ ℕ)       (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾)))

Theorembccn0 40125 Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶C𝑐0) = 1)

Theorembccn1 40126 Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶C𝑐1) = 𝐶)

Theorembccbc 40127 The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾))

20.32.7  Binomial series

Theoremuzmptshftfval 40128* When 𝐹 is a maps-to function on some set of upper integers 𝑍 that returns a set 𝐵, (𝐹 shift 𝑁) is another maps-to function on the shifted set of upper integers 𝑊. (Contributed by Steve Rodriguez, 22-Apr-2020.)
𝐹 = (𝑥𝑍𝐵)    &   𝐵 ∈ V    &   (𝑥 = (𝑦𝑁) → 𝐵 = 𝐶)    &   𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝑁))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝐹 shift 𝑁) = (𝑦𝑊𝐶))

Theoremdvradcnv2 40129* The radius of convergence of the (formal) derivative 𝐻 of the power series 𝐺 is (at least) as large as the radius of convergence of 𝐺. This version of dvradcnv 24728 uses a shifted version of 𝐻 to match the sum form of (ℂ D 𝐹) in pserdv2 24737 (and shows how to use uzmptshftfval 40128 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐻 = (𝑛 ∈ ℕ ↦ ((𝑛 · (𝐴𝑛)) · (𝑋↑(𝑛 − 1))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (abs‘𝑋) < 𝑅)       (𝜑 → seq1( + , 𝐻) ∈ dom ⇝ )

Theorembinomcxplemwb 40130 Lemma for binomcxp 40139. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ)       (𝜑 → (((𝐶𝐾) · (𝐶C𝑐𝐾)) + ((𝐶 − (𝐾 − 1)) · (𝐶C𝑐(𝐾 − 1)))) = (𝐶 · (𝐶C𝑐𝐾)))

Theorembinomcxplemnn0 40131* Lemma for binomcxp 40139. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15044 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set (0...𝐶), and when the index set is widened beyond 𝐶 the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)       ((𝜑𝐶 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴𝑐(𝐶𝑘)) · (𝐵𝑘))))

Theorembinomcxplemrat 40132* Lemma for binomcxp 40139. As 𝑘 increases, this ratio's absolute value converges to one. Part of equation "Since continuity of the absolute value..." in the Wikibooks proof (proven for the inverse ratio, which we later show is no problem). (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐶𝑘) / (𝑘 + 1)))) ⇝ 1)

Theorembinomcxplemfrat 40133* Lemma for binomcxp 40139. binomcxplemrat 40132 implies that when 𝐶 is not a nonnegative integer, the absolute value of the ratio ((𝐹‘(𝑘 + 1)) / (𝐹𝑘)) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))       ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹𝑘)))) ⇝ 1)

Theorembinomcxplemradcnv 40134* Lemma for binomcxp 40139. By binomcxplemfrat 40133 and radcnvrat 40096 the radius of convergence of power series Σ𝑘 ∈ ℕ0((𝐹𝑘) · (𝑏𝑘)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹𝑘) · (𝑏𝑘))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆𝑟)) ∈ dom ⇝ }, ℝ*, < )       ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = 1)

Theorembinomcxplemdvbinom 40135* Lemma for binomcxp 40139. By the power and chain rules, calculate the derivative of ((1 + 𝑏)↑𝑐-𝐶), with respect to 𝑏 in the disk of convergence 𝐷. We later multiply the derivative in the later binomcxplemdvsum 40137 by this derivative to show that ((1 + 𝑏)↑𝑐𝐶) (with a non-negated 𝐶) and the later sum, since both at 𝑏 = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹𝑘) · (𝑏𝑘))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹𝑘)) · (𝑏↑(𝑘 − 1)))))    &   𝐷 = (abs “ (0[,)𝑅))       ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (ℂ D (𝑏𝐷 ↦ ((1 + 𝑏)↑𝑐-𝐶))) = (𝑏𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑𝑐(-𝐶 − 1)))))

Theorembinomcxplemcvg 40136* Lemma for binomcxp 40139. The sum in binomcxplemnn0 40131 and its derivative (see the next theorem, binomcxplemdvsum 40137) converge, as long as their base 𝐽 is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹𝑘) · (𝑏𝑘))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹𝑘)) · (𝑏↑(𝑘 − 1)))))    &   𝐷 = (abs “ (0[,)𝑅))       ((𝜑𝐽𝐷) → (seq0( + , (𝑆𝐽)) ∈ dom ⇝ ∧ seq1( + , (𝐸𝐽)) ∈ dom ⇝ ))

Theorembinomcxplemdvsum 40137* Lemma for binomcxp 40139. The derivative of the generalized sum in binomcxplemnn0 40131. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹𝑘) · (𝑏𝑘))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹𝑘)) · (𝑏↑(𝑘 − 1)))))    &   𝐷 = (abs “ (0[,)𝑅))    &   𝑃 = (𝑏𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆𝑏)‘𝑘))       (𝜑 → (ℂ D 𝑃) = (𝑏𝐷 ↦ Σ𝑘 ∈ ℕ ((𝐸𝑏)‘𝑘)))

Theorembinomcxplemnotnn0 40138* Lemma for binomcxp 40139. When 𝐶 is not a nonnegative integer, the generalized sum in binomcxplemnn0 40131 —which we will call 𝑃 —is a convergent power series: its base 𝑏 is always of smaller absolute value than the radius of convergence.

pserdv2 24737 gives the derivative of 𝑃, which by dvradcnv 24728 also converges in that radius. When 𝐴 is fixed at one, (𝐴 + 𝑏) times that derivative equals (𝐶 · 𝑃) and fraction (𝑃 / ((𝐴 + 𝑏)↑𝑐𝐶)) is always defined with derivative zero, so the fraction is a constant—specifically one, because ((1 + 0)↑𝑐𝐶) = 1. Thus ((1 + 𝑏)↑𝑐𝐶) = (𝑃𝑏).

Finally, let 𝑏 be (𝐵 / 𝐴), and multiply both the binomial ((1 + (𝐵 / 𝐴))↑𝑐𝐶) and the sum (𝑃‘(𝐵 / 𝐴)) by (𝐴𝑐𝐶) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)

(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹𝑘) · (𝑏𝑘))))    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹𝑘)) · (𝑏↑(𝑘 − 1)))))    &   𝐷 = (abs “ (0[,)𝑅))    &   𝑃 = (𝑏𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆𝑏)‘𝑘))       ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴𝑐(𝐶𝑘)) · (𝐵𝑘))))

Theorembinomcxp 40139* Generalize the binomial theorem binom 15044 to positive real summand 𝐴, real summand 𝐵, and complex exponent 𝐶. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus; see also https://en.wikipedia.org/wiki/Binomial_series, https://en.wikipedia.org/wiki/Binomial_theorem (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴𝑐(𝐶𝑘)) · (𝐵𝑘))))

20.33  Mathbox for Andrew Salmon

20.33.1  Principia Mathematica * 10

Theorempm10.12 40140* 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.)
(∀𝑥(𝜑𝜓) → (𝜑 ∨ ∀𝑥𝜓))

Theorempm10.14 40141 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theorempm10.251 40142 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Theorempm10.252 40143 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
(¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)

Theorempm10.253 40144 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑)

Theoremalbitr 40145 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))

Theorempm10.42 40146 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓))

Theorempm10.52 40147* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ 𝜓))

Theorempm10.53 40148 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))

Theorempm10.541 40149* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑𝜓)))

Theorempm10.542 40150* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 → ∀𝑥(𝜑𝜓)))

Theorempm10.55 40151 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))

Theorempm10.56 40152 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜓𝜒))

Theorempm10.57 40153 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ∨ ∃𝑥(𝜑𝜒)))

20.33.2  Principia Mathematica * 11

Theorem2alanimi 40154 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∀𝑥𝑦𝜒)

Theorem2al2imi 40155 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝑦𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))

Theorempm11.11 40156 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
𝜑       𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑

Theorempm11.12 40157* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
(∀𝑥𝑦(𝜑𝜓) → (𝜑 ∨ ∀𝑥𝑦𝜓))

Theorem19.21vv 40158* Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1899. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∀𝑥𝑦𝜑))

Theorem2alim 40159 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))

Theorem2albi 40160 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))

Theorem2exim 40161 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))

Theorem2exbi 40162 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Theoremspsbce-2 40163 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥𝑦𝜑)

Theorem19.33-2 40164 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥𝑦𝜑 ∨ ∀𝑥𝑦𝜓) → ∀𝑥𝑦(𝜑𝜓))

Theorem19.36vv 40165* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))

Theorem19.31vv 40166* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))

Theorem19.37vv 40167* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∃𝑥𝑦𝜑))

Theorem19.28vv 40168* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 ∧ ∀𝑥𝑦𝜑))

Theorempm11.52 40169 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ ¬ ∀𝑥𝑦(𝜑 → ¬ 𝜓))

Theorem2exanali 40170 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremaaanv 40171* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2274. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))

Theorempm11.57 40172* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Theorempm11.58 40173* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝜑 ↔ ∃𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))

Theorempm11.59 40174* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
(∀𝑥(𝜑𝜓) → ∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))

Theorempm11.6 40175* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
(∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))

Theorempm11.61 40176* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑦𝑥(𝜑𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))

Theorempm11.62 40177* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦((𝜑𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓𝜒)))

Theorempm11.63 40178 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 → ∀𝑥𝑦(𝜑𝜓))

Theorempm11.7 40179 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜑𝜑) ↔ ∃𝑥𝑦𝜑)

Theorempm11.71 40180* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑𝜓) ∧ ∀𝑦(𝜒𝜃)) ↔ ∀𝑥𝑦((𝜑𝜒) → (𝜓𝜃))))

20.33.3  Predicate Calculus

Theoremsbeqal1 40181* If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.)
(∀𝑥(𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)

Theoremsbeqal1i 40182* Suppose you know 𝑥 = 𝑦 implies 𝑥 = 𝑧, assuming 𝑥 and 𝑧 are distinct. Then, 𝑦 = 𝑧. (Contributed by Andrew Salmon, 3-Jun-2011.)
(𝑥 = 𝑦𝑥 = 𝑧)       𝑦 = 𝑧

Theoremsbeqal2i 40183* If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.)
(𝑥 = 𝑦𝑥 = 𝑧)       𝑧 = 𝑦

Theoremaxc5c4c711 40184 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1759 as the inference rule. This proof extends the idea of axc5c711 35532 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
((∀𝑥𝑦 ¬ ∀𝑥𝑦(∀𝑦𝜑𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))

Theoremaxc5c4c711toc5 40185 Rederivation of sp 2112 from axc5c4c711 40184. Note that ax6 2315 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1930 instead of ax6 2315, so that this rederivation requires only ax6v 1930 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c4c711toc4 40186 Rederivation of axc4 2262 from axc5c4c711 40184. Note that only propositional calculus is required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremaxc5c4c711toc7 40187 Rederivation of axc7 2258 from axc5c4c711 40184. Note that neither axc7 2258 nor ax-11 2094 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc5c4c711to11 40188 Rederivation of ax-11 2094 from axc5c4c711 40184. Note that ax-11 2094 is not required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremaxc11next 40189* This theorem shows that, given axext4 2749, we can derive a version of axc11n 2363. However, it is weaker than axc11n 2363 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)

20.33.4  Principia Mathematica * 13 and * 14

Theorempm13.13a 40190 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.)
((𝜑𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)

Theorempm13.13b 40191 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
(([𝐴 / 𝑥]𝜑𝑥 = 𝐴) → 𝜑)

Theorempm13.14 40192 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
(([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥𝐴)

Theorempm13.192 40193* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
(∃𝑦(∀𝑥(𝑥 = 𝐴𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Theorempm13.193 40194 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝑥 = 𝑦))

Theorempm13.194 40195 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝜑𝑥 = 𝑦))

Theorempm13.195 40196* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3701. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
(∃𝑦(𝑦 = 𝐴𝜑) ↔ [𝐴 / 𝑦]𝜑)

Theorempm13.196a 40197* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦𝑥))

Theorem2sbc6g 40198* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴𝐶𝐵𝐷) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Theorem2sbc5g 40199* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴𝐶𝐵𝐷) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))

Theoremiotain 40200 Equivalence between two different forms of . (Contributed by Andrew Salmon, 15-Jul-2011.)
(∃!𝑥𝜑 {𝑥𝜑} = (℩𝑥𝜑))

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