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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | expanduniss 41001* | Expand ∪ 𝐴 ⊆ 𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) | ||
Theorem | ismnuprim 41002* | Express the predicate on 𝑈 in ismnu 40969 using only primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (∀𝑧 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑧(𝑧 ∈ 𝑈 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))) | ||
Theorem | rr-grothprimbi 41003* | 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 41008. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥 ¬ ∀𝑦(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))) | ||
Theorem | inagrud 41004 | Inaccessible levels of the cumulative hierarchy are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝜑 → 𝐼 ∈ Inacc) ⇒ ⊢ (𝜑 → (𝑅1‘𝐼) ∈ Univ) | ||
Theorem | inaex 41005* | Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝐴 ∈ On → ∃𝑥 ∈ Inacc 𝐴 ∈ 𝑥) | ||
Theorem | gruex 41006* | Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ ∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 | ||
Theorem | rr-groth 41007* | An equivalent of ax-groth 10234 using only simple defined symbols. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∀𝑓∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑦 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))) | ||
Theorem | rr-grothprim 41008* | 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(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑦 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑦 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))) | ||
Theorem | nanorxor 41009 | '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 41010 | Union of two disjoint restricted class abstractions; compare unrab 4226. (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 41011 | 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 41012 | ℝ is a subset of both ℝ and ℂ. (Contributed by Steve Rodriguez, 22-Nov-2015.) |
⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆) | ||
Theorem | seff 41013 | 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 41014 | The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 23004. (Contributed by Steve Rodriguez, 8-Nov-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) | ||
Theorem | prmunb2 41015* | The primes are unbounded. This generalizes prmunb 16240 to real 𝐴 with arch 11882 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) | ||
Theorem | dvgrat 41016* | 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 41017* |
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 15231 and dvgrat 41016 directly uses the limit of the ratio.
(It also demonstrates how to use climi2 14860 and absltd 14781 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 14781, and how to use r19.29a 3248 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3243 at https://groups.google.com/g/metamath/c/2RPikOiXLMo 3243.) (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 41018* | Let 𝐿 be the limit, if one exists, of the ratio (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) (as in the ratio test cvgdvgrat 41017) 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 41017 —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)) | ||
Theorem | reldvds 41019 | The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
⊢ Rel ∥ | ||
Theorem | nznngen 41020 | 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 41021 | 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 41022 | 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 41023 | 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 41024 | Special case of hashdvds 16102: 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 41025 | Special case of hashnzfz 41024: 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 41026* | As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 41024 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)) | ||
Theorem | caofcan 41027* | Transfer a cancellation law like mulcan 11266 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 41028 | Function analogue of subid 10894. (Contributed by Steve Rodriguez, 5-Nov-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∘f − 𝐹) = (𝐴 × {0})) | ||
Theorem | ofmul12 41029 | Function analogue of mul12 10794. (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 41030 | Function analogue of divrec 11303, a division analogue of ofnegsub 11623. (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 41031 | Function analogue of divcan4 11314. (Contributed by Steve Rodriguez, 4-Nov-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶(ℂ ∖ {0})) → ((𝐹 ∘f · 𝐺) ∘f / 𝐺) = 𝐹) | ||
Theorem | ofdivdiv2 41032 | Function analogue of divdiv2 11341. (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)) | ||
Theorem | lhe4.4ex1a 41033 | Example of the Fundamental Theorem of Calculus, part two (ftc2 24647): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 24647 as simply the "Fundamental Theorem of Calculus", then ftc1 24645 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 41034 | 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 41035 | 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 41036 | 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 41037* | Exponential growth and decay model. See expgrowth 41039 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 41038* | The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 24520 and dveq0 24603. 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 41039* |
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 41037 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 41037); 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 41037 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 41037. Statements for this and expgrowthi 41037 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)))))) | ||
Syntax | cbcc 41040 | Extend class notation to include the generalized binomial coefficient operation. |
class C𝑐 | ||
Definition | df-bcc 41041* | Define a generalized binomial coefficient operation, which unlike df-bc 13659 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 41042 | 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 41043 | Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ) | ||
Theorem | bcc0 41044 | 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 41045 | 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 41046 | 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 41047 | Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐶C𝑐0) = 1) | ||
Theorem | bccn1 41048 | Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐶C𝑐1) = 𝐶) | ||
Theorem | bccbc 41049 | 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)) | ||
Theorem | uzmptshftfval 41050* | 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 41051* | 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 25016 uses a shifted version of 𝐻 to match the sum form of (ℂ D 𝐹) in pserdv2 25025 (and shows how to use uzmptshftfval 41050 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 41052 | Lemma for binomcxp 41061. 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 41053* | Lemma for binomcxp 41061. When 𝐶 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15177 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 41054* | Lemma for binomcxp 41061. 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 41055* | Lemma for binomcxp 41061. binomcxplemrat 41054 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 41056* | Lemma for binomcxp 41061. By binomcxplemfrat 41055 and radcnvrat 41018 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 41057* | Lemma for binomcxp 41061. 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 41059 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 41058* | Lemma for binomcxp 41061. The sum in binomcxplemnn0 41053 and its derivative (see the next theorem, binomcxplemdvsum 41059) 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 41059* | Lemma for binomcxp 41061. The derivative of the generalized sum in binomcxplemnn0 41053. 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)) & ⊢ 𝑃 = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ0 ((𝑆‘𝑏)‘𝑘)) ⇒ ⊢ (𝜑 → (ℂ D 𝑃) = (𝑏 ∈ 𝐷 ↦ Σ𝑘 ∈ ℕ ((𝐸‘𝑏)‘𝑘))) | ||
Theorem | binomcxplemnotnn0 41060* |
Lemma for binomcxp 41061. When 𝐶 is not a nonnegative integer, the
generalized sum in binomcxplemnn0 41053 —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 25025 gives the derivative of 𝑃, which by dvradcnv 25016 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 41061* | Generalize the binomial theorem binom 15177 to positive real summand 𝐴, real summand 𝐵, and complex exponent 𝐶. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15177; see also https://en.wikipedia.org/wiki/Binomial_series 15177, https://en.wikipedia.org/wiki/Binomial_theorem 15177 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15177. (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)))) | ||
Theorem | pm10.12 41062* | 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 41063 | 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 41064 | Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | ||
Theorem | pm10.252 41065 | Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) |
⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑) | ||
Theorem | pm10.253 41066 | Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ (¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑) | ||
Theorem | albitr 41067 | 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 41068 | Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | pm10.52 41069* | Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓)) | ||
Theorem | pm10.53 41070 | Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | pm10.541 41071* | 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 41072* | 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 41073 | 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 41074 | 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 41075 | Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒))) | ||
Theorem | 2alanimi 41076 | 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 41077 | 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 41078 | 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 41079* | Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓)) | ||
Theorem | 19.21vv 41080* | Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1940. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑)) | ||
Theorem | 2alim 41081 | 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 41082 | 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 41083 | 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 41084 | 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 41085 | 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 41086 | 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 41087* | 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 41088* | 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 41089* | 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 41090* | 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 41091 | Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓)) | ||
Theorem | aaanv 41092* | Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2342. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) | ||
Theorem | pm11.57 41093* | Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | pm11.58 41094* | Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | pm11.59 41095* | 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 41096* | 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 41097* | Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓)) | ||
Theorem | pm11.62 41098* | 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 41099 | Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | pm11.7 41100 | Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
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