HomeHome Metamath Proof Explorer
Theorem List (p. 406 of 449)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28623)
  Hilbert Space Explorer  Hilbert Space Explorer
(28624-30146)
  Users' Mathboxes  Users' Mathboxes
(30147-44804)
 

Theorem List for Metamath Proof Explorer - 40501-40600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrumnudlem 40501* Lemma for grumnud 40502. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝐺 ∈ Univ)    &   𝐹 = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))    &   ((𝑖𝐺𝐺) → (𝑖𝐹 ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))    &   (( ∈ (𝐹 Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → ∃𝑢𝑓 (𝑖𝑢 𝑢 ∈ (𝐹 Coll 𝑧)))       (𝜑𝐺𝑀)
 
Theoremgrumnud 40502* Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.)
𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}    &   (𝜑𝐺 ∈ Univ)       (𝜑𝐺𝑀)
 
Theoremgrumnueq 40503* The class of Grothendieck universes is equal to the class of minimal universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Univ = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
 
20.32.2.3  Primitive equivalent of ax-groth
 
Theoremexpandan 40504 Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ ¬ (𝜓 → ¬ 𝜃))
 
Theoremexpandexn 40505 Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑 ↔ ¬ 𝜓)       (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓)
 
Theoremexpandral 40506 Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜓))
 
Theoremexpandrexn 40507 Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑 ↔ ¬ 𝜓)       (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴𝜓))
 
Theoremexpandrex 40508 Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴 → ¬ 𝜓))
 
Theoremexpanduniss 40509* Expand 𝐴𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
( 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
 
Theoremismnuprim 40510* Express the predicate on 𝑈 in ismnu 40477 using only primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))) ↔ ∀𝑧(𝑧𝑈 → ∀𝑓 ¬ ∀𝑤(𝑤𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑈 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑈 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤))))))))))))
 
Theoremrr-grothprimbi 40511* Express "every set is contained in a Grothendieck universe" using only primitives. The right side (without the outermost universal quantifier) is proven as rr-grothprim 40516. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(∀𝑥𝑦 ∈ Univ 𝑥𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤)))))))))))))
 
Theoreminagrud 40512 Inaccessible levels of the cumulative hierarchy are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝜑𝐼 ∈ Inacc)       (𝜑 → (𝑅1𝐼) ∈ Univ)
 
Theoreminaex 40513* Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴𝑥)
 
Theoremgruex 40514* Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑦 ∈ Univ 𝑥𝑦
 
Theoremrr-groth 40515* An equivalent of ax-groth 10234 using only simple defined symbols. (Contributed by Rohan Ridenour, 13-Aug-2023.)
𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∀𝑓𝑤𝑦 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑦 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
 
Theoremrr-grothprim 40516* An equivalent of ax-groth 10234 using only primitives. This uses only 123 symbols, which is significantly less than the previous record of 163 established by grothprim 10245 (which uses some defined symbols, and requires 229 symbols if expanded to primitives). (Contributed by Rohan Ridenour, 13-Aug-2023.)
¬ ∀𝑦(𝑥𝑦 → ¬ ∀𝑧(𝑧𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡𝑣𝑡𝑧) → ¬ (𝑣𝑦 → ¬ 𝑣𝑤)) → ¬ ∀𝑖(𝑖𝑧 → (𝑣𝑦 → (𝑖𝑣 → (𝑣𝑓 → ¬ ∀𝑢(𝑢𝑓 → (𝑖𝑢 → ¬ ∀𝑜(𝑜𝑢 → ∀𝑠(𝑠𝑜𝑠𝑤))))))))))))
 
20.33  Mathbox for Steve Rodriguez
 
20.33.1  Miscellanea
 
Theoremnanorxor 40517 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
 
Theoremundisjrab 40518 Union of two disjoint restricted class abstractions; compare unrab 4273. (Contributed by Steve Rodriguez, 28-Feb-2020.)
(({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = ∅ ↔ ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)})
 
Theoremiso0 40519 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 40520 is a subset of both and . (Contributed by Steve Rodriguez, 22-Nov-2015.)
(𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆)
 
Theoremseff 40521 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 40522 The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 22936. (Contributed by Steve Rodriguez, 8-Nov-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆))       ((𝜑𝑃𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆)
 
Theoremprmunb2 40523* The primes are unbounded. This generalizes prmunb 16240 to real 𝐴 with arch 11883 and lttrd 10790: 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)
 
20.33.2  Ratio test for infinite series convergence and divergence
 
Theoremdvgrat 40524* 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 40525* 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 15229 and dvgrat 40524 directly uses the limit of the ratio.

(It also demonstrates how to use climi2 14858 and absltd 14779 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 14779, and how to use r19.29a 3289 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3284 at https://groups.google.com/g/metamath/c/2RPikOiXLMo 3284.) (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 40526* Let 𝐿 be the limit, if one exists, of the ratio (abs‘((𝐴‘(𝑘 + 1)) / (𝐴𝑘))) (as in the ratio test cvgdvgrat 40525) 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 40525 —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))
 
20.33.3  Multiples
 
Theoremreldvds 40527 The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Rel ∥
 
Theoremnznngen 40528 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 40529 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 40530 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 40531 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 40532 Special case of hashdvds 16102: 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 40533 Special case of hashnzfz 40532: 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 40534* As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 40532 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))
 
20.33.4  Function operations
 
Theoremcaofcan 40535* Transfer a cancellation law like mulcan 11266 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 40536 Function analogue of subid 10894. (Contributed by Steve Rodriguez, 5-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ) → (𝐹f𝐹) = (𝐴 × {0}))
 
Theoremofmul12 40537 Function analogue of mul12 10794. (Contributed by Steve Rodriguez, 13-Nov-2015.)
(((𝐴𝑉𝐹:𝐴⟶ℂ) ∧ (𝐺:𝐴⟶ℂ ∧ 𝐻:𝐴⟶ℂ)) → (𝐹f · (𝐺f · 𝐻)) = (𝐺f · (𝐹f · 𝐻)))
 
Theoremofdivrec 40538 Function analogue of divrec 11303, a division analogue of ofnegsub 11625. (Contributed by Steve Rodriguez, 3-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → (𝐹f · ((𝐴 × {1}) ∘f / 𝐺)) = (𝐹f / 𝐺))
 
Theoremofdivcan4 40539 Function analogue of divcan4 11314. (Contributed by Steve Rodriguez, 4-Nov-2015.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → ((𝐹f · 𝐺) ∘f / 𝐺) = 𝐹)
 
Theoremofdivdiv2 40540 Function analogue of divdiv2 11341. (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))
 
20.33.5  Calculus
 
Theoremlhe4.4ex1a 40541 Example of the Fundamental Theorem of Calculus, part two (ftc2 24570): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 24570 as simply the "Fundamental Theorem of Calculus", then ftc1 24568 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 40542 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 40543 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 40544 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 40545* Exponential growth and decay model. See expgrowth 40547 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 40546* The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 24443 and dveq0 24526. 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 40547* 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 40545 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 40545); 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 40545 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 40545.

Statements for this and expgrowthi 40545 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))))))
 
20.33.6  The generalized binomial coefficient operation
 
Syntaxcbcc 40548 Extend class notation to include the generalized binomial coefficient operation.
class C𝑐
 
Definitiondf-bcc 40549* Define a generalized binomial coefficient operation, which unlike df-bc 13653 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘)))
 
Theorembccval 40550 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 40551 Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ)
 
Theorembcc0 40552 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 40553 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 40554 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 40555 Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶C𝑐0) = 1)
 
Theorembccn1 40556 Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐶C𝑐1) = 𝐶)
 
Theorembccbc 40557 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))
 
20.33.7  Binomial series
 
Theoremuzmptshftfval 40558* 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 40559* 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 24938 uses a shifted version of 𝐻 to match the sum form of (ℂ D 𝐹) in pserdv2 24947 (and shows how to use uzmptshftfval 40558 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 40560 Lemma for binomcxp 40569. 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 40561* Lemma for binomcxp 40569. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15175 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 40562* Lemma for binomcxp 40569. 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 40563* Lemma for binomcxp 40569. binomcxplemrat 40562 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 40564* Lemma for binomcxp 40569. By binomcxplemfrat 40563 and radcnvrat 40526 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 40565* Lemma for binomcxp 40569. 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 40567 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.)
(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 40566* Lemma for binomcxp 40569. The sum in binomcxplemnn0 40561 and its derivative (see the next theorem, binomcxplemdvsum 40567) 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 40567* Lemma for binomcxp 40569. The derivative of the generalized sum in binomcxplemnn0 40561. 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 40568* Lemma for binomcxp 40569. When 𝐶 is not a nonnegative integer, the generalized sum in binomcxplemnn0 40561 —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 24947 gives the derivative of 𝑃, which by dvradcnv 24938 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 40569* Generalize the binomial theorem binom 15175 to positive real summand 𝐴, real summand 𝐵, and complex exponent 𝐶. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15175; see also https://en.wikipedia.org/wiki/Binomial_series 15175, https://en.wikipedia.org/wiki/Binomial_theorem 15175 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15175. (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))))
 
20.34  Mathbox for Andrew Salmon
 
20.34.1  Principia Mathematica * 10
 
Theorempm10.12 40570* 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 40571 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
 
Theorempm10.251 40572 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)
 
Theorempm10.252 40573 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
(¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
 
Theorempm10.253 40574 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑)
 
Theoremalbitr 40575 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))
 
Theorempm10.42 40576 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓))
 
Theorempm10.52 40577* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ 𝜓))
 
Theorempm10.53 40578 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝜓))
 
Theorempm10.541 40579* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑𝜓)))
 
Theorempm10.542 40580* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 → ∀𝑥(𝜑𝜓)))
 
Theorempm10.55 40581 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((∃𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) ↔ (∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)))
 
Theorempm10.56 40582 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜓𝜒))
 
Theorempm10.57 40583 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ∨ ∃𝑥(𝜑𝜒)))
 
20.34.2  Principia Mathematica * 11
 
Theorem2alanimi 40584 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∀𝑥𝑦𝜒)
 
Theorem2al2imi 40585 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝑦𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))
 
Theorempm11.11 40586 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
𝜑       𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
 
Theorempm11.12 40587* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
(∀𝑥𝑦(𝜑𝜓) → (𝜑 ∨ ∀𝑥𝑦𝜓))
 
Theorem19.21vv 40588* Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1931. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∀𝑥𝑦𝜑))
 
Theorem2alim 40589 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 40590 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 40591 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 40592 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 40593 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑥𝑦𝜑)
 
Theorem19.33-2 40594 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 40595* 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 40596* 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 40597* 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 40598* 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 40599 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ ¬ ∀𝑥𝑦(𝜑 → ¬ 𝜓))
 
Theoremaaanv 40600* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2345. (Contributed by Andrew Salmon, 24-May-2011.)
((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
  Copyright terms: Public domain < Previous  Next >