![]() |
Metamath
Proof Explorer Theorem List (p. 405 of 437) | < 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: | ![]() (1-28347) |
![]() (28348-29872) |
![]() (29873-43650) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | divlt0gt0d 40401 | The ratio of a negative numerator and a positive denominator is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) < 0) | ||
Theorem | subsub23d 40402 | Swap subtrahend and result of subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵)) | ||
Theorem | 2timesgt 40403 | Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ ℝ+ → 𝐴 < (2 · 𝐴)) | ||
Theorem | reopn 40404 | The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ℝ ∈ (topGen‘ran (,)) | ||
Theorem | elfzop1le2 40405 | A member in a half-open integer interval plus 1 is less than or equal to the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 + 1) ≤ 𝑁) | ||
Theorem | sub31 40406 | Swap the first and third terms in a double subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = (𝐶 − (𝐵 − 𝐴))) | ||
Theorem | nnne1ge2 40407 | A positive integer which is not 1 is greater than or equal to 2. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁) | ||
Theorem | lefldiveq 40408 | A closed enough, smaller real 𝐶 has the same floor of 𝐴 when both are divided by 𝐵. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) ⇒ ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) | ||
Theorem | negsubdi3d 40409 | Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → -(𝐴 − 𝐵) = (-𝐴 − -𝐵)) | ||
Theorem | ltdiv2dd 40410 | Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐶 / 𝐵) < (𝐶 / 𝐴)) | ||
Theorem | abssinbd 40411 | Bound for the absolute value of the sine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴 ∈ ℝ → (abs‘(sin‘𝐴)) ≤ 1) | ||
Theorem | halffl 40412 | Floor of (1 / 2). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (⌊‘(1 / 2)) = 0 | ||
Theorem | monoords 40413* | Ordering relation for a strictly monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1))) & ⊢ (𝜑 → 𝐼 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐼 < 𝐽) ⇒ ⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘𝐽)) | ||
Theorem | hashssle 40414 | The size of a subset of a finite set is less than the size of the containing set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by hashss 13511, and hashssle 40414 should be deleted afterwards. |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (♯‘𝐵) ≤ (♯‘𝐴)) | ||
Theorem | lttri5d 40415 | Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → ¬ 𝐵 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 < 𝐵) | ||
Theorem | fzisoeu 40416* | A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13560 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐻 ∈ Fin) & ⊢ (𝜑 → < Or 𝐻) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑁 = ((♯‘𝐻) + (𝑀 − 1)) ⇒ ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)) | ||
Theorem | lt3addmuld 40417 | If three real numbers are less than a fourth real number, the sum of the three real numbers is less than three times the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐷) & ⊢ (𝜑 → 𝐵 < 𝐷) & ⊢ (𝜑 → 𝐶 < 𝐷) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) < (3 · 𝐷)) | ||
Theorem | absnpncan2d 40418 | Triangular inequality, combined with cancellation law for subtraction (applied twice). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐷)) ≤ (((abs‘(𝐴 − 𝐵)) + (abs‘(𝐵 − 𝐶))) + (abs‘(𝐶 − 𝐷)))) | ||
Theorem | fperiodmullem 40419* | A function with period T is also periodic with period nonnegative multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) | ||
Theorem | fperiodmul 40420* | A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹‘(𝑋 + (𝑁 · 𝑇))) = (𝐹‘𝑋)) | ||
Theorem | upbdrech 40421* | Choice of an upper bound for a nonempty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) & ⊢ 𝐶 = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) ⇒ ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
Theorem | lt4addmuld 40422 | If four real numbers are less than a fifth real number, the sum of the four real numbers is less than four times the fifth real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐸) & ⊢ (𝜑 → 𝐵 < 𝐸) & ⊢ (𝜑 → 𝐶 < 𝐸) & ⊢ (𝜑 → 𝐷 < 𝐸) ⇒ ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) + 𝐷) < (4 · 𝐸)) | ||
Theorem | absnpncan3d 40423 | Triangular inequality, combined with cancellation law for subtraction (applied three times). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ ℂ) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐸)) ≤ ((((abs‘(𝐴 − 𝐵)) + (abs‘(𝐵 − 𝐶))) + (abs‘(𝐶 − 𝐷))) + (abs‘(𝐷 − 𝐸)))) | ||
Theorem | upbdrech2 40424* | Choice of an upper bound for a possibly empty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) & ⊢ 𝐶 = if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) ⇒ ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
Theorem | ssfiunibd 40425* | A finite union of bounded sets is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝑥 𝐵 ≤ 𝑦) & ⊢ (𝜑 → 𝐶 ⊆ ∪ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ 𝐶 𝐵 ≤ 𝑤) | ||
Theorem | fzdifsuc2 40426 | Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12718, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘(𝑀 − 1)) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)})) | ||
Theorem | fzsscn 40427 | A finite sequence of integers is a set of complex numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝑀...𝑁) ⊆ ℂ | ||
Theorem | divcan8d 40428 | A cancellation law for division. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐵 / (𝐴 · 𝐵)) = (1 / 𝐴)) | ||
Theorem | dmmcand 40429 | Cancellation law for division and multiplication. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐵 · 𝐶)) = (𝐴 · 𝐶)) | ||
Theorem | fzssre 40430 | A finite sequence of integers is a set of real numbers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝑀...𝑁) ⊆ ℝ | ||
Theorem | elfzelzd 40431 | A member of a finite set of sequential integer is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → 𝐾 ∈ ℤ) | ||
Theorem | bccld 40432 | A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑁C𝐾) ∈ ℕ0) | ||
Theorem | leadd12dd 40433 | Addition to both sides of 'less than or equal to'. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) | ||
Theorem | fzssnn0 40434 | A finite set of sequential integers that is a subset of ℕ0. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (0...𝑁) ⊆ ℕ0 | ||
Theorem | xreqle 40435 | Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | ||
Theorem | xaddid2d 40436 | 0 is a left identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (0 +𝑒 𝐴) = 𝐴) | ||
Theorem | xadd0ge 40437 | A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐴 +𝑒 𝐵)) | ||
Theorem | elfzolem1 40438 | A member in a half-open integer interval is less than or equal to the upper bound minus 1 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ (𝑁 − 1)) | ||
Theorem | xrgtned 40439 | 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) | ||
Theorem | xrleneltd 40440 | 'Less than or equal to' and 'not equals' implies 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 < 𝐵) | ||
Theorem | xaddcomd 40441 | The extended real addition operation is commutative. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴)) | ||
Theorem | supxrre3 40442* | The supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) | ||
Theorem | uzfissfz 40443* | For any finite subset of the upper integers, there is a finite set of sequential integers that includes it. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ 𝑍 𝐴 ⊆ (𝑀...𝑘)) | ||
Theorem | xleadd2d 40444 | Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵)) | ||
Theorem | suprltrp 40445* | The supremum of a nonempty bounded set of reals can be approximated from below by elements of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧) | ||
Theorem | xleadd1d 40446 | Addition of extended reals preserves the "less than or equal to" relation, in the left slot. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)) | ||
Theorem | xreqled 40447 | Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | xrgepnfd 40448 | An extended real greater than or equal to +∞ is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → +∞ ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = +∞) | ||
Theorem | xrge0nemnfd 40449 | A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → 𝐴 ≠ -∞) | ||
Theorem | supxrgere 40450* | If a real number can be approximated from below by members of a set, then it is less than or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦) ⇒ ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ*, < )) | ||
Theorem | iuneqfzuzlem 40451* | Lemma for iuneqfzuz 40452: here, inclusion is proven; aiuneqfzuz uses this lemma twice, to prove equality. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝑍 = (ℤ≥‘𝑁) ⇒ ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵) | ||
Theorem | iuneqfzuz 40452* | If two unions indexed by upper integers are equal if they agree on any partial indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝑍 = (ℤ≥‘𝑁) ⇒ ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑛 ∈ 𝑍 𝐵) | ||
Theorem | xle2addd 40453 | Adding both side of two inequalities. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐷 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷)) | ||
Theorem | supxrgelem 40454* | If an extended real number can be approximated from below by members of a set, then it is less than or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +𝑒 𝑥)) ⇒ ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ*, < )) | ||
Theorem | supxrge 40455* | If an extended real number can be approximated from below by members of a set, then it is less than or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ 𝐴 𝐵 ≤ (𝑦 +𝑒 𝑥)) ⇒ ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ*, < )) | ||
Theorem | suplesup 40456* | If any element of 𝐴 can be approximated from below by members of 𝐵, then the supremum of 𝐴 is less than or equal to the supremum of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ⊆ ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < )) | ||
Theorem | infxrglb 40457* | The infimum of a set of extended reals is less than an extended real if and only if the set contains a smaller number. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) → (inf(𝐴, ℝ*, < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) | ||
Theorem | xadd0ge2 40458 | A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐵 +𝑒 𝐴)) | ||
Theorem | nepnfltpnf 40459 | An extended real that is not +∞ is less than +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ≠ +∞) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → 𝐴 < +∞) | ||
Theorem | ltadd12dd 40460 | Addition to both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐵 < 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) | ||
Theorem | nemnftgtmnft 40461 | An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → -∞ < 𝐴) | ||
Theorem | xrgtso 40462 | 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ◡ < Or ℝ* | ||
Theorem | rpex 40463 | The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ℝ+ ∈ V | ||
Theorem | xrge0ge0 40464 | A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) | ||
Theorem | xrssre 40465 | A subset of extended reals that does not contain +∞ and -∞ is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → ¬ +∞ ∈ 𝐴) & ⊢ (𝜑 → ¬ -∞ ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ⊆ ℝ) | ||
Theorem | ssuzfz 40466 | A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ⊆ 𝑍) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) | ||
Theorem | absfun 40467 | The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ Fun abs | ||
Theorem | infrpge 40468* | The infimum of a nonempty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ*, < ) +𝑒 𝐵)) | ||
Theorem | xrlexaddrp 40469* | If an extended real number 𝐴 can be approximated from above, adding positive reals to 𝐵, then 𝐴 is less than or equal to 𝐵. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝐴 ≤ (𝐵 +𝑒 𝑥)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | supsubc 40470* | The supremum function distributes over subtraction in a sense similar to that in supaddc 11344. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐶 = {𝑧 ∣ ∃𝑣 ∈ 𝐴 𝑧 = (𝑣 − 𝐵)} ⇒ ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝐵) = sup(𝐶, ℝ, < )) | ||
Theorem | xralrple2 40471* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. A variant on xralrple 12348. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ ((1 + 𝑥) · 𝐵))) | ||
Theorem | nnuzdisj 40472 | The first 𝑁 elements of the set of nonnegative integers are distinct from any later members. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ ((1...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) = ∅ | ||
Theorem | ltdivgt1 40473 | Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → (1 < 𝐵 ↔ (𝐴 / 𝐵) < 𝐴)) | ||
Theorem | xrltned 40474 | 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | nnsplit 40475 | Express the set of positive integers as the disjoint (see nnuzdisj 40472) union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ (𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) | ||
Theorem | divdiv3d 40476 | Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐶 · 𝐵))) | ||
Theorem | abslt2sqd 40477 | Comparison of the square of two numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐴) < (abs‘𝐵)) ⇒ ⊢ (𝜑 → (𝐴↑2) < (𝐵↑2)) | ||
Theorem | qenom 40478 | The set of rational numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ ℚ ≈ ω | ||
Theorem | qct 40479 | The set of rational numbers is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ ℚ ≼ ω | ||
Theorem | xrltnled 40480 | 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
Theorem | lenlteq 40481 | 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → ¬ 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | xrred 40482 | An extended real that is neither minus infinity, nor plus infinity, is real. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≠ -∞) & ⊢ (𝜑 → 𝐴 ≠ +∞) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | rr2sscn2 40483 | The cartesian square of ℝ is a subset of the cartesian square of ℂ. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ) | ||
Theorem | infxr 40484* | The infimum of a set of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝐵 < 𝑥 → ∃𝑦 ∈ 𝐴 𝑦 < 𝑥)) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) = 𝐵) | ||
Theorem | infxrunb2 40485* | The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ inf(𝐴, ℝ*, < ) = -∞)) | ||
Theorem | infxrbnd2 40486* | The infimum of a bounded-below set of extended reals is greater than minus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ -∞ < inf(𝐴, ℝ*, < ))) | ||
Theorem | infleinflem1 40487 | Lemma for infleinf 40489, case 𝐵 ≠ ∅ ∧ -∞ < inf(𝐵, ℝ*, < ). (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ⊆ ℝ*) & ⊢ (𝜑 → 𝑊 ∈ ℝ+) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≤ (inf(𝐵, ℝ*, < ) +𝑒 (𝑊 / 2))) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑍 ≤ (𝑋 +𝑒 (𝑊 / 2))) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ (inf(𝐵, ℝ*, < ) +𝑒 𝑊)) | ||
Theorem | infleinflem2 40488 | Lemma for infleinf 40489, when inf(𝐵, ℝ*, < ) = -∞. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ⊆ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 < (𝑅 − 2)) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑍 ≤ (𝑋 +𝑒 1)) ⇒ ⊢ (𝜑 → 𝑍 < 𝑅) | ||
Theorem | infleinf 40489* | If any element of 𝐵 can be approximated from above by members of 𝐴, then the infimum of 𝐴 is less than or equal to the infimum of 𝐵. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ⊆ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +𝑒 𝑦)) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < )) | ||
Theorem | xralrple4 40490* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝑥↑𝑁)))) | ||
Theorem | xralrple3 40491* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + (𝐶 · 𝑥)))) | ||
Theorem | eluzelzd 40492 | A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → 𝑁 ∈ ℤ) | ||
Theorem | suplesup2 40493* | If any element of 𝐴 is less than or equal to an element in 𝐵, then the supremum of 𝐴 is less than or equal to the supremum of 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ⊆ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < )) | ||
Theorem | recnnltrp 40494 | 𝑁 is a natural number large enough that its reciprocal is smaller than the given positive 𝐸. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1) ⇒ ⊢ (𝐸 ∈ ℝ+ → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸)) | ||
Theorem | fiminre2 40495* | A nonempty finite set of real numbers is bounded below. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | ||
Theorem | nnn0 40496 | The set of positive integers is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ ℕ ≠ ∅ | ||
Theorem | fzct 40497 | A finite set of sequential integer is countable. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝑁...𝑀) ≼ ω | ||
Theorem | rpgtrecnn 40498* | Any positive real number is greater than the reciprocal of a positive integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝐴 ∈ ℝ+ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐴) | ||
Theorem | fzossuz 40499 | A half-open integer interval is a subset of an upper set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝑀..^𝑁) ⊆ (ℤ≥‘𝑀) | ||
Theorem | fzossz 40500 | A half-open integer interval is a set of integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝑀..^𝑁) ⊆ ℤ |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |