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Theorem List for Metamath Proof Explorer - 37401-37500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremint-ineqtransd 37401 InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐵)       (𝜑𝐶𝐴)
 
20.27.3  N-Digit Addition Proof Generator

This section formalizes theorems used in an n-digit addition proof generator.

Other theorems required: deccl 11251 addcomli 9978 00id 9961 addid1i 9973 addid2i 9974 eqid 2514 dec0h 11261 decadd 11309 decaddc 11311.

 
Theoremunitadd 37402 Theorem used in conjunction with decaddc 11311 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝐴 + 𝐵) = 𝐹    &   (𝐶 + 1) = 𝐵    &   𝐴 ∈ ℕ0    &   𝐶 ∈ ℕ0       ((𝐴 + 𝐶) + 1) = 𝐹
 
20.27.4  AM-GM (for k = 2,3,4)
 
Theoremgsumws3 37403 Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑆𝐵 ∧ (𝑇𝐵𝑈𝐵))) → (𝐺 Σg ⟨“𝑆𝑇𝑈”⟩) = (𝑆 + (𝑇 + 𝑈)))
 
Theoremgsumws4 37404 Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ (𝑆𝐵 ∧ (𝑇𝐵 ∧ (𝑈𝐵𝑉𝐵)))) → (𝐺 Σg ⟨“𝑆𝑇𝑈𝑉”⟩) = (𝑆 + (𝑇 + (𝑈 + 𝑉))))
 
Theoremamgm2d 37405 Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 24404. (Contributed by Stanislas Polu, 8-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2))
 
Theoremamgm3d 37406 Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3))
 
Theoremamgm4d 37407 Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)       (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4))
 
20.28  Mathbox for Steve Rodriguez
 
20.28.1  Miscellanea
 
Theoremnanorxor 37408 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
 
Theoremundisjrab 37409 Union of two disjoint restricted class abstractions; compare unrab 3760. (Contributed by Steve Rodriguez, 28-Feb-2020.)
(({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = ∅ ↔ ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)})
 
Theoremiso0 37410 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 (∅, ∅)
 
Theoremssrecnpr 37411 is a subset of both and . (Contributed by Steve Rodriguez, 22-Nov-2015.)
(𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)
 
Theoremseff 37412 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.)
(𝜑𝑆 ∈ {ℝ, ℂ})       (𝜑 → (exp ↾ 𝑆):𝑆𝑆)
 
Theoremsblpnf 37413 The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 21912. (Contributed by Steve Rodriguez, 8-Nov-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆))       ((𝜑𝑃𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆)
 
Theoremprmunb2 37414* The primes are unbounded. This generalizes prmunb 15338 to real 𝐴 with arch 11043 and lttrd 9948: 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)
 
20.28.2  Ratio test for infinite series convergence and divergence
 
Theoremdvgrat 37415* Ratio test for divergence of a complex infinite series. See e.g. remark "if (abs‘((𝑎‘(𝑛 + 1)) / (𝑎𝑛))) ≥ 1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑊) → (𝐹𝑘) ≠ 0)    &   ((𝜑𝑘𝑊) → (abs‘(𝐹𝑘)) ≤ (abs‘(𝐹‘(𝑘 + 1))))       (𝜑 → seq𝑀( + , 𝐹) ∉ dom ⇝ )
 
Theoremcvgdvgrat 37416* 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 14321 and dvgrat 37415 directly uses the limit of the ratio.

(It also demonstrates how to use climi2 13954 and absltd 13873 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191, and how to use r19.29a 2964 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 2917 at https://groups.google.com/forum/#!topic/metamath/2RPikOiXLMo.) (Contributed by Steve Rodriguez, 28-Feb-2020.)

𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑊) → (𝐹𝑘) ≠ 0)    &   𝑅 = (𝑘𝑊 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹𝑘))))    &   (𝜑𝑅𝐿)    &   (𝜑𝐿 ≠ 1)       (𝜑 → (𝐿 < 1 ↔ seq𝑀( + , 𝐹) ∈ dom ⇝ ))
 
Theoremradcnvrat 37417* Let 𝐿 be the limit, if one exists, of the ratio (abs‘((𝐴‘(𝑘 + 1)) / (𝐴𝑘))) (as in the ratio test cvgdvgrat 37416) as 𝑘 increases. Then the radius of convergence of power series Σ𝑛 ∈ ℕ0((𝐴𝑛) · (𝑥𝑛)) is (1 / 𝐿) if 𝐿 is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence —a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐷 = (𝑘 ∈ ℕ0 ↦ (abs‘((𝐴‘(𝑘 + 1)) / (𝐴𝑘))))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℕ0)    &   ((𝜑𝑘𝑍) → (𝐴𝑘) ≠ 0)    &   (𝜑𝐷𝐿)    &   (𝜑𝐿 ≠ 0)       (𝜑𝑅 = (1 / 𝐿))
 
20.28.3  Multiples
 
Theoremreldvds 37418 The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Rel ∥
 
Theoremnznngen 37419 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.)
(𝜑𝑁 ∈ ℤ)       (𝜑 → (( ∥ “ {𝑁}) ∩ ℕ) ⊆ (ℤ‘(abs‘𝑁)))
 
Theoremnzss 37420 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.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁𝑉)       (𝜑 → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁𝑀))
 
Theoremnzin 37421 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.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
 
Theoremnzprmdif 37422 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 37423 Special case of hashdvds 15194: 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 37424 Special case of hashnzfz 37423: 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 37425* As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 37423 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))
 
20.28.4  Function operations
 
Theoremcaofcan 37426* Transfer a cancellation law like mulcan 10412 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 37427 Function analogue of subid 10050. (Contributed by Steve Rodriguez, 5-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ) → (𝐹𝑓𝐹) = (𝐴 × {0}))
 
Theoremofmul12 37428 Function analogue of mul12 9952. (Contributed by Steve Rodriguez, 13-Nov-2015.)
(((𝐴𝑉𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶ℂ ∧ 𝐻:𝐴⟶ℂ)) → (𝐹𝑓 · (𝐺𝑓 · 𝐻)) = (𝐺𝑓 · (𝐹𝑓 · 𝐻)))
 
Theoremofdivrec 37429 Function analogue of divrec 10449, a division analogue of ofnegsub 10772. (Contributed by Steve Rodriguez, 3-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → (𝐹𝑓 · ((𝐴 × {1}) ∘𝑓 / 𝐺)) = (𝐹𝑓 / 𝐺))
 
Theoremofdivcan4 37430 Function analogue of divcan4 10460. (Contributed by Steve Rodriguez, 4-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → ((𝐹𝑓 · 𝐺) ∘𝑓 / 𝐺) = 𝐹)
 
Theoremofdivdiv2 37431 Function analogue of divdiv2 10485. (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))
 
20.28.5  Calculus
 
Theoremlhe4.4ex1a 37432 Example of the Fundamental Theorem of Calculus, part two (ftc2 23487): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 23487 as simply the "Fundamental Theorem of Calculus", then ftc1 23485 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 37433 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 37434 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 37435 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 37436* Exponential growth and decay model. See expgrowth 37438 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 37437* The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 23362 and dveq0 23443. 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 37438* 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 37436 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 37436 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))))))
 
20.28.6  The generalized binomial coefficient operation
 
Syntaxcbcc 37439 Extend class notation to include the generalized binomial coefficient operation.
class C𝑐
 
Definitiondf-bcc 37440* Define a generalized binomial coefficient operation, which unlike df-bc 12819 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
 
Theorembccval 37441 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 37442 Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ)
 
Theorembcc0 37443 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 37444 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 37445 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 37446 Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶C𝑐0) = 1)
 
Theorembccn1 37447 Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶C𝑐1) = 𝐶)
 
Theorembccbc 37448 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))
 
20.28.7  Binomial series
 
Theoremuzmptshftfval 37449* 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 37450* 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 23867 uses a shifted version of 𝐻 to match the sum form of (ℂ D 𝐹) in pserdv2 23876 (and shows how to use uzmptshftfval 37449 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 37451 Lemma for binomcxp 37460. 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 37452* Lemma for binomcxp 37460. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 14268 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 37453* Lemma for binomcxp 37460. 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 37454* Lemma for binomcxp 37460. binomcxplemrat 37453 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 37455* Lemma for binomcxp 37460. By binomcxplemfrat 37454 and radcnvrat 37417 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 37456* Lemma for binomcxp 37460. 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 37458 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 37457* Lemma for binomcxp 37460. The sum in binomcxplemnn0 37452 and its derivative (see the next theorem, binomcxplemdvsum 37458) 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 37458* Lemma for binomcxp 37460. The derivative of the generalized sum in binomcxplemnn0 37452. 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 37459* Lemma for binomcxp 37460. When 𝐶 is not a nonnegative integer, the generalized sum in binomcxplemnn0 37452 —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 23876 gives the derivative of 𝑃, which by dvradcnv 23867 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 37460* Generalize the binomial theorem binom 14268 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))))
 
20.29  Mathbox for Andrew Salmon
 
20.29.1  Principia Mathematica * 10
 
Theorempm10.12 37461* 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 37462 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
 
Theorempm10.251 37463 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
 
Theorempm10.252 37464 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
(¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
 
Theorempm10.253 37465 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑)
 
Theoremalbitr 37466 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))
 
Theorempm10.42 37467 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓))
 
Theorempm10.52 37468* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ 𝜓))
 
Theorempm10.53 37469 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
 
Theorempm10.541 37470* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑𝜓)))
 
Theorempm10.542 37471* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 → ∀𝑥(𝜑𝜓)))
 
Theorempm10.55 37472 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))
 
Theorempm10.56 37473 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜓𝜒))
 
Theorempm10.57 37474 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ∨ ∃𝑥(𝜑𝜒)))
 
20.29.2  Principia Mathematica * 11
 
Theorem2alanimi 37475 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∀𝑥𝑦𝜒)
 
Theorem2al2imi 37476 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝑦𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))
 
Theorempm11.11 37477 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
𝜑       𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
 
Theorempm11.12 37478* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
(∀𝑥𝑦(𝜑𝜓) → (𝜑 ∨ ∀𝑥𝑦𝜓))
 
Theorem19.21vv 37479* Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1821. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∀𝑥𝑦𝜑))
 
Theorem2alim 37480 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 37481 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 37482 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 37483 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 37484 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥𝑦𝜑)
 
Theorem19.33-2 37485 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 37486* 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 37487* 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 37488* 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 37489* 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 37490 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ ¬ ∀𝑥𝑦(𝜑 → ¬ 𝜓))
 
Theorem2exanali 37491 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
 
Theoremaaanv 37492* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2106. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
 
Theorempm11.57 37493* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
 
Theorempm11.58 37494* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝜑 ↔ ∃𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
 
Theorempm11.59 37495* 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 37496* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
(∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))
 
Theorempm11.61 37497* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑦𝑥(𝜑𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
 
Theorempm11.62 37498* 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 37499 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 → ∀𝑥𝑦(𝜑𝜓))
 
Theorempm11.7 37500 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜑𝜑) ↔ ∃𝑥𝑦𝜑)
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