P. 444 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44301-44400   *Has distinct variable group(s)

Type	Label	Description
Statement
21.39  Mathbox for Steve Rodriguez
21.39.1  Miscellanea
Theorem	nanorxor 44301	'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ ((𝜑 ⊼ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ⊻ 𝜓)))
Theorem	undisjrab 44302	Union of two disjoint restricted class abstractions; compare unrab 4281. (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ∅ ↔ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ⊻ 𝜓)})
Theorem	iso0 44303	The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
⊢ ∅ Isom 𝑅, 𝑆 (∅, ∅)
Theorem	ssrecnpr 44304	ℝ is a subset of both ℝ and ℂ. (Contributed by Steve Rodriguez, 22-Nov-2015.)
⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)
Theorem	seff 44305	Let set 𝑆 be the real or complex numbers. Then the exponential function restricted to 𝑆 is a mapping from 𝑆 to 𝑆. (Contributed by Steve Rodriguez, 6-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    ⇒   ⊢ (𝜑 → (exp ↾ 𝑆):𝑆⟶𝑆)
Theorem	sblpnf 44306	The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 24292. (Contributed by Steve Rodriguez, 8-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆))    ⇒   ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆)
Theorem	prmunb2 44307*	The primes are unbounded. This generalizes prmunb 16892 to real 𝐴 with arch 12446 and lttrd 11342: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝐴 ∈ ℝ → ∃𝑝 ∈ ℙ 𝐴 < 𝑝)
21.39.2  Ratio test for infinite series convergence and divergence
Theorem	dvgrat 44308*	Ratio test for divergence of a complex infinite series. See e.g. remark "if (abs‘((𝑎‘(𝑛 + 1)) / (𝑎‘𝑛))) ≥ 1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ≤ (abs‘(𝐹‘(𝑘 + 1))))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∉ dom ⇝ )
Theorem	cvgdvgrat 44309*	Ratio test for convergence and divergence of a complex infinite series. If the ratio 𝑅 of the absolute values of successive terms in an infinite sequence 𝐹 converges to less than one, then the infinite sum of the terms of 𝐹 converges to a complex number; and if 𝑅 converges greater then the sum diverges. This combined form of cvgrat 15856 and dvgrat 44308 directly uses the limit of the ratio. (It also demonstrates how to use climi2 15484 and absltd 15405 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 15405, and how to use r19.29a 3142 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3135 at https://groups.google.com/g/metamath/c/2RPikOiXLMo 3135.) (Contributed by Steve Rodriguez, 28-Feb-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ≠ 0)    &   ⊢ 𝑅 = (𝑘 ∈ 𝑊 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘))))    &   ⊢ (𝜑 → 𝑅 ⇝ 𝐿)    &   ⊢ (𝜑 → 𝐿 ≠ 1)    ⇒   ⊢ (𝜑 → (𝐿 < 1 ↔ seq𝑀( + , 𝐹) ∈ dom ⇝ ))
Theorem	radcnvrat 44310*	Let 𝐿 be the limit, if one exists, of the ratio (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) (as in the ratio test cvgdvgrat 44309) as 𝑘 increases. Then the radius of convergence of power series Σ𝑛 ∈ ℕ0((𝐴‘𝑛) · (𝑥↑𝑛)) is (1 / 𝐿) if 𝐿 is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence 44309 —a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐷 = (𝑘 ∈ ℕ0 ↦ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴‘𝑘) ≠ 0)    &   ⊢ (𝜑 → 𝐷 ⇝ 𝐿)    &   ⊢ (𝜑 → 𝐿 ≠ 0)    ⇒   ⊢ (𝜑 → 𝑅 = (1 / 𝐿))
21.39.3  Multiples
Theorem	reldvds 44311	The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ Rel ∥
Theorem	nznngen 44312	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.)
⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑁}) ∩ ℕ) ⊆ (ℤ≥‘(abs‘𝑁)))
Theorem	nzss 44313	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀))
Theorem	nzin 44314	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
Theorem	nzprmdif 44315	Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑀 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℙ)    &   ⊢ (𝜑 → 𝑀 ≠ 𝑁)    ⇒   ⊢ (𝜑 → (( ∥ “ {𝑀}) ∖ ( ∥ “ {𝑁})) = (( ∥ “ {𝑀}) ∖ ( ∥ “ {(𝑀 · 𝑁)})))
Theorem	hashnzfz 44316	Special case of hashdvds 16752: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘(𝐽 − 1)))    ⇒   ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (𝐽...𝐾))) = ((⌊‘(𝐾 / 𝑁)) − (⌊‘((𝐽 − 1) / 𝑁))))
Theorem	hashnzfz2 44317	Special case of hashnzfz 44316: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁)))
Theorem	hashnzfzclim 44318*	As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 44316 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.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) ⇝ (1 / 𝑀))
21.39.4  Function operations
Theorem	caofcan 44319*	Transfer a cancellation law like mulcan 11822 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑇)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦) = (𝑥𝑅𝑧) ↔ 𝑦 = 𝑧))    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f 𝑅𝐻) ↔ 𝐺 = 𝐻))
Theorem	ofsubid 44320	Function analogue of subid 11448. (Contributed by Steve Rodriguez, 5-Nov-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∘f − 𝐹) = (𝐴 × {0}))
Theorem	ofmul12 44321	Function analogue of mul12 11346. (Contributed by Steve Rodriguez, 13-Nov-2015.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶ℂ ∧ 𝐻:𝐴⟶ℂ)) → (𝐹 ∘f · (𝐺 ∘f · 𝐻)) = (𝐺 ∘f · (𝐹 ∘f · 𝐻)))
Theorem	ofdivrec 44322	Function analogue of divrec 11860, a division analogue of ofnegsub 12191. (Contributed by Steve Rodriguez, 3-Nov-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → (𝐹 ∘f · ((𝐴 × {1}) ∘f / 𝐺)) = (𝐹 ∘f / 𝐺))
Theorem	ofdivcan4 44323	Function analogue of divcan4 11871. (Contributed by Steve Rodriguez, 4-Nov-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → ((𝐹 ∘f · 𝐺) ∘f / 𝐺) = 𝐹)
Theorem	ofdivdiv2 44324	Function analogue of divdiv2 11901. (Contributed by Steve Rodriguez, 23-Nov-2015.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶(ℂ ∖ {0}) ∧ 𝐻:𝐴⟶(ℂ ∖ {0}))) → (𝐹 ∘f / (𝐺 ∘f / 𝐻)) = ((𝐹 ∘f · 𝐻) ∘f / 𝐺))
21.39.5  Calculus
Theorem	lhe4.4ex1a 44325	Example of the Fundamental Theorem of Calculus, part two (ftc2 25958): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 25958 as simply the "Fundamental Theorem of Calculus", then ftc1 25956 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)
Theorem	dvsconst 44326	Derivative of a constant function on the real or complex numbers. The function may return a complex 𝐴 even if 𝑆 is ℝ. (Contributed by Steve Rodriguez, 11-Nov-2015.)
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐴 ∈ ℂ) → (𝑆 D (𝑆 × {𝐴})) = (𝑆 × {0}))
Theorem	dvsid 44327	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}))
Theorem	dvsef 44328	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))
Theorem	expgrowthi 44329*	Exponential growth and decay model. See expgrowth 44331 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐾 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝑌 = (𝑡 ∈ 𝑆 ↦ (𝐶 · (exp‘(𝐾 · 𝑡))))    ⇒   ⊢ (𝜑 → (𝑆 D 𝑌) = ((𝑆 × {𝐾}) ∘f · 𝑌))
Theorem	dvconstbi 44330*	The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 25825 and dveq0 25912. Corresponds to integration formula "∫0 d𝑥 = 𝐶 " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑌:𝑆⟶ℂ)    &   ⊢ (𝜑 → dom (𝑆 D 𝑌) = 𝑆)    ⇒   ⊢ (𝜑 → ((𝑆 D 𝑌) = (𝑆 × {0}) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑆 × {𝑐})))
Theorem	expgrowth 44331*	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 44329 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 44329); 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 44329 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 44329. Statements for this and expgrowthi 44329 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐾 ∈ ℂ)    &   ⊢ (𝜑 → 𝑌:𝑆⟶ℂ)    &   ⊢ (𝜑 → dom (𝑆 D 𝑌) = 𝑆)    ⇒   ⊢ (𝜑 → ((𝑆 D 𝑌) = ((𝑆 × {𝐾}) ∘f · 𝑌) ↔ ∃𝑐 ∈ ℂ 𝑌 = (𝑡 ∈ 𝑆 ↦ (𝑐 · (exp‘(𝐾 · 𝑡))))))
21.39.6  The generalized binomial coefficient operation
Syntax	cbcc 44332	Extend class notation to include the generalized binomial coefficient operation.
class C𝑐
Definition	df-bcc 44333*	Define a generalized binomial coefficient operation, which unlike df-bc 14275 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
Theorem	bccval 44334	Value of the generalized binomial coefficient, 𝐶 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐶C𝑐𝐾) = ((𝐶 FallFac 𝐾) / (!‘𝐾)))
Theorem	bcccl 44335	Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ)
Theorem	bcc0 44336	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.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐶C𝑐𝐾) = 0 ↔ 𝐶 ∈ (0...(𝐾 − 1))))
Theorem	bccp1k 44337	Generalized binomial coefficient: 𝐶 choose (𝐾 + 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐶C𝑐(𝐾 + 1)) = ((𝐶C𝑐𝐾) · ((𝐶 − 𝐾) / (𝐾 + 1))))
Theorem	bccm1k 44338	Generalized binomial coefficient: 𝐶 choose (𝐾 − 1), when 𝐶 is not (𝐾 − 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ (ℂ ∖ {(𝐾 − 1)}))    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐶C𝑐(𝐾 − 1)) = ((𝐶C𝑐𝐾) / ((𝐶 − (𝐾 − 1)) / 𝐾)))
Theorem	bccn0 44339	Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶C𝑐0) = 1)
Theorem	bccn1 44340	Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶C𝑐1) = 𝐶)
Theorem	bccbc 44341	The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑁C𝑐𝐾) = (𝑁C𝐾))
21.39.7  Binomial series
Theorem	uzmptshftfval 44342*	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 = (𝑥 ∈ 𝑍 ↦ 𝐵)    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = (𝑦 − 𝑁) → 𝐵 = 𝐶)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝐹 shift 𝑁) = (𝑦 ∈ 𝑊 ↦ 𝐶))
Theorem	dvradcnv2 44343*	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 26337 uses a shifted version of 𝐻 to match the sum form of (ℂ D 𝐹) in pserdv2 26347 (and shows how to use uzmptshftfval 44342 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝑛 · (𝐴‘𝑛)) · (𝑋↑(𝑛 − 1))))    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝑋) < 𝑅)    ⇒   ⊢ (𝜑 → seq1( + , 𝐻) ∈ dom ⇝ )
Theorem	binomcxplemwb 44344	Lemma for binomcxp 44353. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (((𝐶 − 𝐾) · (𝐶C𝑐𝐾)) + ((𝐶 − (𝐾 − 1)) · (𝐶C𝑐(𝐾 − 1)))) = (𝐶 · (𝐶C𝑐𝐾)))
Theorem	binomcxplemnn0 44345*	Lemma for binomcxp 44353. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15803 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.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴↑𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
Theorem	binomcxplemrat 44346*	Lemma for binomcxp 44353. 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.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐶 − 𝑘) / (𝑘 + 1)))) ⇝ 1)
Theorem	binomcxplemfrat 44347*	Lemma for binomcxp 44353. binomcxplemrat 44346 implies that when 𝐶 is not a nonnegative integer, the absolute value of the ratio ((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)))) ⇝ 1)
Theorem	binomcxplemradcnv 44348*	Lemma for binomcxp 44353. By binomcxplemfrat 44347 and radcnvrat 44310 the radius of convergence of power series Σ𝑘 ∈ ℕ0((𝐹‘𝑘) · (𝑏↑𝑘)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = 1)
Theorem	binomcxplemdvbinom 44349*	Lemma for binomcxp 44353. 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 44351 by this derivative to show that ((1 + 𝑏)↑𝑐𝐶) (with a nonnegated 𝐶) and the later sum, since both at 𝑏 = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (ℂ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑𝑐-𝐶))) = (𝑏 ∈ 𝐷 ↦ (-𝐶 · ((1 + 𝑏)↑𝑐(-𝐶 − 1)))))
Theorem	binomcxplemcvg 44350*	Lemma for binomcxp 44353. The sum in binomcxplemnn0 44345 and its derivative (see the next theorem, binomcxplemdvsum 44351) 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.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (seq0( + , (𝑆‘𝐽)) ∈ dom ⇝ ∧ seq1( + , (𝐸‘𝐽)) ∈ dom ⇝ ))
Theorem	binomcxplemdvsum 44351*	Lemma for binomcxp 44353. The derivative of the generalized sum in binomcxplemnn0 44345. Part of remark "This convergence allows to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    &   ⊢ 𝑃 = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘))    ⇒   ⊢ (𝜑 → (ℂ D 𝑃) = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘)))
Theorem	binomcxplemnotnn0 44352*	Lemma for binomcxp 44353. When 𝐶 is not a nonnegative integer, the generalized sum in binomcxplemnn0 44345 —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 26347 gives the derivative of 𝑃, which by dvradcnv 26337 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) = (𝑃‘𝑏). Finally, let 𝑏 be (𝐵 / 𝐴), and multiply both the binomial ((1 + (𝐵 / 𝐴))↑𝑐𝐶) and the sum (𝑃‘(𝐵 / 𝐴)) by (𝐴↑𝑐𝐶) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))    &   ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))))    &   ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))    &   ⊢ 𝐷 = (◡abs “ (0[,)𝑅))    &   ⊢ 𝑃 = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴↑𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
Theorem	binomcxp 44353*	Generalize the binomial theorem binom 15803 to positive real summand 𝐴, real summand 𝐵, and complex exponent 𝐶. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15803; see also https://en.wikipedia.org/wiki/Binomial_series 15803, https://en.wikipedia.org/wiki/Binomial_theorem 15803 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15803. (Contributed by Steve Rodriguez, 22-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) = Σ𝑘 ∈ ℕ0 ((𝐶C𝑐𝑘) · ((𝐴↑𝑐(𝐶 − 𝑘)) · (𝐵↑𝑘))))
21.40  Mathbox for Andrew Salmon
21.40.1  Principia Mathematica * 10
Theorem	pm10.12 44354*	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(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥𝜓))
Theorem	pm10.14 44355	Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Theorem	pm10.251 44356	Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
Theorem	pm10.252 44357	Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
Theorem	pm10.253 44358	Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑)
Theorem	albitr 44359	Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥(𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜓 ↔ 𝜒)) → ∀𝑥(𝜑 ↔ 𝜒))
Theorem	pm10.42 44360	Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∨ 𝜓))
Theorem	pm10.52 44361*	Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓))
Theorem	pm10.53 44362	Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))
Theorem	pm10.541 44363*	Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜒 ∨ 𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.542 44364*	Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.55 44365	Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∃𝑥(𝜑 ∧ 𝜓) ∧ ∀𝑥(𝜑 → 𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)))
Theorem	pm10.56 44366	Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒))
Theorem	pm10.57 44367	Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒)))
21.40.2  Principia Mathematica * 11
Theorem	2alanimi 44368	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦𝜒)
Theorem	2al2imi 44369	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒))
Theorem	pm11.11 44370	Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ 𝜑    ⇒   ⊢ ∀𝑧∀𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
Theorem	pm11.12 44371*	Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))
Theorem	19.21vv 44372*	Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1939. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑))
Theorem	2alim 44373	Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	2albi 44374	Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	2exim 44375	Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	2exbi 44376	Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	spsbce-2 44377	Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	19.33-2 44378	Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))
Theorem	19.36vv 44379*	Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓))
Theorem	19.31vv 44380*	Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓))
Theorem	19.37vv 44381*	Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))
Theorem	19.28vv 44382*	Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∀𝑥∀𝑦𝜑))
Theorem	pm11.52 44383	Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓))
Theorem	aaanv 44384*	Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2331. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))
Theorem	pm11.57 44385*	Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Theorem	pm11.58 44386*	Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Theorem	pm11.59 44387*	Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
Theorem	pm11.6 44388*	Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	pm11.61 44389*	Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
Theorem	pm11.62 44390*	Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓 → 𝜒)))
Theorem	pm11.63 44391	Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))
Theorem	pm11.7 44392	Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
Theorem	pm11.71 44393*	Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜒) → ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑦(𝜒 → 𝜃)) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))))
21.40.3  Predicate Calculus
Theorem	sbeqal1 44394*	If 𝑥 = 𝑦 always implies 𝑥 = 𝑧, then 𝑦 = 𝑧. (Contributed by Andrew Salmon, 2-Jun-2011.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑧) → 𝑦 = 𝑧)
Theorem	sbeqal1i 44395*	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 = 𝑦 → 𝑥 = 𝑧)    ⇒   ⊢ 𝑦 = 𝑧
Theorem	sbeqal2i 44396*	If 𝑥 = 𝑦 implies 𝑥 = 𝑧, then we can infer 𝑧 = 𝑦. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑧)    ⇒   ⊢ 𝑧 = 𝑦
Theorem	axc5c4c711 44397	Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1795 as the inference rule. This proof extends the idea of axc5c711 38918 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
⊢ ((∀𝑥∀𝑦 ¬ ∀𝑥∀𝑦(∀𝑦𝜑 → 𝜓) → (𝜑 → ∀𝑦(∀𝑦𝜑 → 𝜓))) → (∀𝑦𝜑 → ∀𝑦𝜓))
Theorem	axc5c4c711toc5 44398	Rederivation of sp 2184 from axc5c4c711 44397. Note that ax6 2383 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1968 instead of ax6 2383, so that this rederivation requires only ax6v 1968 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	axc5c4c711toc4 44399	Rederivation of axc4 2320 from axc5c4c711 44397. 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(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	axc5c4c711toc7 44400	Rederivation of axc7 2316 from axc5c4c711 44397. Note that neither axc7 2316 nor ax-11 2158 are required for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)