Home | Metamath
Proof Explorer Theorem List (p. 419 of 450) | < 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
(1-28697) |
Hilbert Space Explorer
(28698-30220) |
Users' Mathboxes
(30221-44913) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iccshift 41801* | A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) | ||
Theorem | iccsuble 41802 | An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴)) | ||
Theorem | iocopn 41803 | A left-open right-closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝐽 = (𝐾 ↾t (𝐴(,]𝐵)) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐶(,]𝐵) ∈ 𝐽) | ||
Theorem | eliccelioc 41804 | Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴))) | ||
Theorem | iooshift 41805* | An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) | ||
Theorem | iccintsng 41806 | Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵}) | ||
Theorem | icoiccdif 41807 | Left-closed right-open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ((𝐴[,]𝐵) ∖ {𝐵})) | ||
Theorem | icoopn 41808 | A left-closed right-open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝐽 = (𝐾 ↾t (𝐴[,)𝐵)) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴[,)𝐶) ∈ 𝐽) | ||
Theorem | icoub 41809 | A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵)) | ||
Theorem | eliccxrd 41810 | Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | ||
Theorem | pnfel0pnf 41811 | +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ +∞ ∈ (0[,]+∞) | ||
Theorem | eliccnelico 41812 | An element of a closed interval that is not a member of the left-closed right-open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 = 𝐵) | ||
Theorem | eliccelicod 41813 | A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) | ||
Theorem | ge0xrre 41814 | A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) | ||
Theorem | ge0lere 41815 | A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ∈ ℝ) | ||
Theorem | elicores 41816* | Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦)) | ||
Theorem | inficc 41817 | The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝑆 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑆 ≠ ∅) ⇒ ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) | ||
Theorem | qinioo 41818 | The rational numbers are dense in ℝ. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | lenelioc 41819 | A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ≤ 𝐴) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) | ||
Theorem | ioonct 41820 | A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝐶 = (𝐴(,)𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ≼ ω) | ||
Theorem | xrgtnelicc 41821 | A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) | ||
Theorem | iccdificc 41822 | The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((𝐴[,]𝐶) ∖ (𝐴[,]𝐵)) = (𝐵(,]𝐶)) | ||
Theorem | iocnct 41823 | A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝐶 = (𝐴(,]𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ≼ ω) | ||
Theorem | iccnct 41824 | A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝐶 = (𝐴[,]𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ≼ ω) | ||
Theorem | iooiinicc 41825* | A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ ((𝐴 − (1 / 𝑛))(,)(𝐵 + (1 / 𝑛))) = (𝐴[,]𝐵)) | ||
Theorem | iccgelbd 41826 | An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
Theorem | iooltubd 41827 | An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 < 𝐵) | ||
Theorem | icoltubd 41828 | An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 < 𝐵) | ||
Theorem | qelioo 41829* | The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵)) | ||
Theorem | tgqioo2 41830* | Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) ⇒ ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞)) | ||
Theorem | iccleubd 41831 | An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
Theorem | elioored 41832 | A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵(,)𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | ioogtlbd 41833 | An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | ioofun 41834 | (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Fun (,) | ||
Theorem | icomnfinre 41835 | A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) | ||
Theorem | sqrlearg 41836 | The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴↑2) ≤ 𝐴 ↔ 𝐴 ∈ (0[,]1))) | ||
Theorem | ressiocsup 41837 | If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (-∞(,]𝑆) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
Theorem | ressioosup 41838 | If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (-∞(,)𝑆) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
Theorem | iooiinioc 41839* | A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵)) | ||
Theorem | ressiooinf 41840 | If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = inf(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (𝑆(,)+∞) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
Theorem | icogelbd 41841 | An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
Theorem | iocleubd 41842 | An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
Theorem | uzinico 41843 | An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) | ||
Theorem | preimaiocmnf 41844* | Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (-∞(,]𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≤ 𝐵}) | ||
Theorem | uzinico2 41845 | An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) | ||
Theorem | uzinico3 41846 | An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) | ||
Theorem | icossico2 41847 | Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶)) | ||
Theorem | dmico 41848 | The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ dom [,) = (ℝ* × ℝ*) | ||
Theorem | ndmico 41849 | The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ∅) | ||
Theorem | uzubioo 41850* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | uzubico 41851* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | uzubioo2 41852* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | uzubico2 41853* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | iocgtlbd 41854 | An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | xrtgioo2 41855 | The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ) | ||
Theorem | tgioo4 41856 | The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) | ||
Theorem | fsumclf 41857* | Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsumcl 15092 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℂ) | ||
Theorem | fsummulc1f 41858* | Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 15142 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) | ||
Theorem | fsumnncl 41859* | Closure of a nonempty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ) | ||
Theorem | fsumsplit1 41860* | Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐷 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) | ||
Theorem | fsumge0cl 41861* | The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) | ||
Theorem | fsumf1of 41862* | Re-index a finite sum using a bijection. Same as fsumf1o 15082, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷) | ||
Theorem | fsumiunss 41863* | Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 15178, but here 𝐴 and 𝐵 need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ Fin) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐷)𝐶 = Σ𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐷) ≠ ∅}Σ𝑘 ∈ (𝐵 ∩ 𝐷)𝐶) | ||
Theorem | fsumreclf 41864* | Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) | ||
Theorem | fsumlessf 41865* | A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | fsumsupp0 41866* | Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹‘𝑘) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘)) | ||
Theorem | fsumsermpt 41867* | A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴) & ⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | fmul01 41868* | Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ 𝐴 = seq𝐿( · , 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) & ⊢ (𝜑 → 𝐾 ∈ (𝐿...𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) ⇒ ⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1)) | ||
Theorem | fmulcl 41869* | If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑁) & ⊢ (𝜑 → 𝑁 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) & ⊢ (𝜑 → 𝑇 ∈ V) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑌) | ||
Theorem | fmuldfeqlem1 41870* | induction step for the proof of fmuldfeq 41871. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑓𝜑 & ⊢ Ⅎ𝑔𝜑 & ⊢ Ⅎ𝑡𝑌 & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) & ⊢ (𝜑 → 𝑁 ∈ (1...𝑀)) & ⊢ (𝜑 → (𝑁 + 1) ∈ (1...𝑀)) & ⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑁)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1))) | ||
Theorem | fmuldfeq 41871* | X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑡𝑌 & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡)) | ||
Theorem | fmul01lt1lem1 41872* | Given a finite multiplication of values betweeen 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ 𝐴 = seq𝐿( · , 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸) ⇒ ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) | ||
Theorem | fmul01lt1lem2 41873* | Given a finite multiplication of values betweeen 0 and 1, a value 𝐸 larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ 𝐴 = seq𝐿( · , 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀)) & ⊢ (𝜑 → (𝐵‘𝐽) < 𝐸) ⇒ ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) | ||
Theorem | fmul01lt1 41874* | Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑖𝐵 & ⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ 𝐴 = seq1( · , 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸) ⇒ ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸) | ||
Theorem | cncfmptss 41875* | A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) | ||
Theorem | rrpsscn 41876 | The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ ℝ+ ⊆ ℂ | ||
Theorem | mulc1cncfg 41877* | A version of mulc1cncf 23515 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ)) | ||
Theorem | infrglb 41878* | The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵)) | ||
Theorem | expcnfg 41879* | If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 23532. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ)) | ||
Theorem | prodeq2ad 41880* | Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶) | ||
Theorem | fprodsplit1 41881* | Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) | ||
Theorem | fprodexp 41882* | Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁)) | ||
Theorem | fprodabs2 41883* | The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)) | ||
Theorem | fprod0 41884* | A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐶 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 = 0) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) | ||
Theorem | mccllem 41885* | * Induction step for mccl 41886. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) & ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m (𝐶 ∪ {𝐷}))) & ⊢ (𝜑 → ∀𝑏 ∈ (ℕ0 ↑m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ) ⇒ ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ) | ||
Theorem | mccl 41886* | A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ Ⅎ𝑘𝐵 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m 𝐴)) ⇒ ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐴 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐴 (!‘(𝐵‘𝑘))) ∈ ℕ) | ||
Theorem | fprodcnlem 41887* | A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾)) | ||
Theorem | fprodcn 41888* | A finite product of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) | ||
Theorem | clim1fr1 41889* | A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴 · 𝑛) + 𝐵) / (𝐴 · 𝑛))) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 1) | ||
Theorem | isumneg 41890* | Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = -Σ𝑘 ∈ 𝑍 𝐴) | ||
Theorem | climrec 41891* | Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺 ⇝ 𝐴) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) | ||
Theorem | climmulf 41892* | A version of climmul 14991 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ⇝ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) | ||
Theorem | climexp 41893* | The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐻 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁)) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁)) | ||
Theorem | climinf 41894* | A bounded monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) | ||
Theorem | climsuselem1 41895* | The subsequence index 𝐼 has the expected properties: it belongs to the same upper integers as the original index, and it is always greater than or equal to the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ≥‘((𝐼‘𝑘) + 1))) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐼‘𝐾) ∈ (ℤ≥‘𝐾)) | ||
Theorem | climsuse 41896* | A subsequence 𝐺 of a converging sequence 𝐹, converges to the same limit. 𝐼 is the strictly increasing and it is used to index the subsequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐼 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ≥‘((𝐼‘𝑘) + 1))) & ⊢ (𝜑 → 𝐺 ∈ 𝑌) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘))) ⇒ ⊢ (𝜑 → 𝐺 ⇝ 𝐴) | ||
Theorem | climrecf 41897* | A version of climrec 41891 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐺 ⇝ 𝐴) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) & ⊢ (𝜑 → 𝐻 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) | ||
Theorem | climneg 41898* | Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -𝐴) | ||
Theorem | climinff 41899* | A version of climinf 41894 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < )) | ||
Theorem | climdivf 41900* | Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ Ⅎ𝑘𝐻 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ⇝ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |