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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | grumnudlem 40501* | Lemma for grumnud 40502. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝐺 ∈ Univ) & ⊢ 𝐹 = ({〈𝑏, 𝑐〉 ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) & ⊢ ((𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺) → (𝑖𝐹ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))) & ⊢ ((ℎ ∈ (𝐹 Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (𝐹 Coll 𝑧))) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝑀) | ||
Theorem | grumnud 40502* | Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} & ⊢ (𝜑 → 𝐺 ∈ Univ) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝑀) | ||
Theorem | grumnueq 40503* | The class of Grothendieck universes is equal to the class of minimal universes. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ Univ = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} | ||
Theorem | expandan 40504 | Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒) ↔ ¬ (𝜓 → ¬ 𝜃)) | ||
Theorem | expandexn 40505 | Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝜑 ↔ ¬ 𝜓) ⇒ ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓) | ||
Theorem | expandral 40506 | Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | ||
Theorem | expandrexn 40507 | Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝜑 ↔ ¬ 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | ||
Theorem | expandrex 40508 | Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜓)) | ||
Theorem | expanduniss 40509* | Expand ∪ 𝐴 ⊆ 𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) | ||
Theorem | ismnuprim 40510* | Express the predicate on 𝑈 in ismnu 40477 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 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.) |
⊢ (∀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 40512 | 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 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 𝐴 ∈ 𝑥) | ||
Theorem | gruex 40514* | 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 40515* | 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 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(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 40517 | '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 40518 | Union of two disjoint restricted class abstractions; compare unrab 4273. (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 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 (∅, ∅) | ||
Theorem | ssrecnpr 40520 | ℝ is a subset of both ℝ and ℂ. (Contributed by Steve Rodriguez, 22-Nov-2015.) |
⊢ (𝑆 ∈ {ℝ, ℂ} → ℝ ⊆ 𝑆) | ||
Theorem | seff 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.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) ⇒ ⊢ (𝜑 → (exp ↾ 𝑆):𝑆⟶𝑆) | ||
Theorem | sblpnf 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.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) | ||
Theorem | prmunb2 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) | ||
Theorem | dvgrat 40524* | 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 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 = (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 40526* | Let 𝐿 be the limit, if one exists, of the ratio (abs‘((𝐴‘(𝑘 + 1)) / (𝐴‘𝑘))) (as in the ratio test cvgdvgrat 40525) 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 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 ↦ ((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 40527 | The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
⊢ Rel ∥ | ||
Theorem | nznngen 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.) |
⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (( ∥ “ {𝑁}) ∩ ℕ) ⊆ (ℤ≥‘(abs‘𝑁))) | ||
Theorem | nzss 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ 𝑉) ⇒ ⊢ (𝜑 → (( ∥ “ {𝑀}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑀)) | ||
Theorem | nzin 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)})) | ||
Theorem | nzprmdif 40531 | 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 40532 | 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 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.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐾 ∈ ℕ) ⇒ ⊢ (𝜑 → (♯‘(( ∥ “ {𝑁}) ∩ (2...𝐾))) = (⌊‘(𝐾 / 𝑁))) | ||
Theorem | hashnzfzclim 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.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐽 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑘 ∈ (ℤ≥‘(𝐽 − 1)) ↦ ((♯‘(( ∥ “ {𝑀}) ∩ (𝐽...𝑘))) / 𝑘)) ⇝ (1 / 𝑀)) | ||
Theorem | caofcan 40535* | 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 40536 | Function analogue of subid 10894. (Contributed by Steve Rodriguez, 5-Nov-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ) → (𝐹 ∘f − 𝐹) = (𝐴 × {0})) | ||
Theorem | ofmul12 40537 | 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 40538 | Function analogue of divrec 11303, a division analogue of ofnegsub 11625. (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 40539 | 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 40540 | 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 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) | ||
Theorem | dvsconst 40542 | 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 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})) | ||
Theorem | dvsef 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)) | ||
Theorem | expgrowthi 40545* | Exponential growth and decay model. See expgrowth 40547 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 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.) |
⊢ (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 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.) |
⊢ (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 40548 | Extend class notation to include the generalized binomial coefficient operation. |
class C𝑐 | ||
Definition | df-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))) | ||
Theorem | bccval 40550 | 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 40551 | Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐶C𝑐𝐾) ∈ ℂ) | ||
Theorem | bcc0 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.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐶C𝑐𝐾) = 0 ↔ 𝐶 ∈ (0...(𝐾 − 1)))) | ||
Theorem | bccp1k 40553 | 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 40554 | 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 40555 | Generalized binomial coefficient: 𝐶 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐶C𝑐0) = 1) | ||
Theorem | bccn1 40556 | Generalized binomial coefficient: 𝐶 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐶C𝑐1) = 𝐶) | ||
Theorem | bccbc 40557 | 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 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 = (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 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 ↦ ((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 40560 | Lemma for binomcxp 40569. 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 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.) |
⊢ (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 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.) |
⊢ (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 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)) / (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 40564* | Lemma for binomcxp 40569. By binomcxplemfrat 40563 and radcnvrat 40526 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 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.) |
⊢ (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 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.) |
⊢ (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 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.) |
⊢ (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 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) = (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 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.) |
⊢ (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 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(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥𝜓)) | ||
Theorem | pm10.14 40571 | 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 40572 | Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | ||
Theorem | pm10.252 40573 | Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) |
⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑) | ||
Theorem | pm10.253 40574 | Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ (¬ ∀𝑥𝜑 ↔ ∃𝑥 ¬ 𝜑) | ||
Theorem | albitr 40575 | 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 40576 | Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | pm10.52 40577* | Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ 𝜓)) | ||
Theorem | pm10.53 40578 | Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | pm10.541 40579* | 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 40580* | 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 40581 | 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 40582 | 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 40583 | 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 40584 | 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 40585 | 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 40586 | 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 40587* | Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓)) | ||
Theorem | 19.21vv 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.) |
⊢ (∀𝑥∀𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∀𝑥∀𝑦𝜑)) | ||
Theorem | 2alim 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.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | 2albi 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.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)) | ||
Theorem | 2exim 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.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)) | ||
Theorem | 2exbi 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.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)) | ||
Theorem | spsbce-2 40593 | 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 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.) |
⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓)) | ||
Theorem | 19.36vv 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.) |
⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓)) | ||
Theorem | 19.31vv 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.) |
⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∨ 𝜓)) | ||
Theorem | 19.37vv 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.) |
⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑)) | ||
Theorem | 19.28vv 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.) |
⊢ (∀𝑥∀𝑦(𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ∀𝑥∀𝑦𝜑)) | ||
Theorem | pm11.52 40599 | Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓)) | ||
Theorem | aaanv 40600* | Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2345. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) |
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