HomeHome Metamath Proof Explorer
Theorem List (p. 473 of 491)
< Previous  Next >
Bad symbols? Try the
GIF version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30946)
  Hilbert Space Explorer  Hilbert Space Explorer
(30947-32469)
  Users' Mathboxes  Users' Mathboxes
(32470-49035)
 

Theorem List for Metamath Proof Explorer - 47201-47300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfatsnafv2 47201 Singleton of function value, analogous to fnsnfv 6987. (Contributed by AV, 7-Sep-2022.)
(𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
 
Theoremdfafv23 47202* A definition of function value in terms of iota, analogous to dffv3 6902. (Contributed by AV, 6-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
 
Theoremdfatdmfcoafv2 47203 Domain of a function composition, analogous to dmfco 7004. (Contributed by AV, 7-Sep-2022.)
(𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))
 
Theoremdfatcolem 47204* Lemma for dfatco 47205. (Contributed by AV, 8-Sep-2022.)
((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹𝐺)𝑦)
 
Theoremdfatco 47205 The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.)
((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → (𝐹𝐺) defAt 𝑋)
 
Theoremafv2co2 47206 Value of a function composition, analogous to fvco2 7005. (Contributed by AV, 8-Sep-2022.)
((𝐺 defAt 𝑋𝐹 defAt (𝐺''''𝑋)) → ((𝐹𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
 
Theoremrlimdmafv2 47207 Two ways to express that a function has a limit, analogous to rlimdm 15583. (Contributed by AV, 5-Sep-2022.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟 ''''𝐹)))
 
Theoremdfafv22 47208 Alternate definition of (𝐹''''𝐴) using (𝐹𝐴) directly. (Contributed by AV, 3-Sep-2022.)
(𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
 
Theoremafv2ndeffv0 47209 If the alternate function value at an argument is undefined, i.e., not in the range of the function, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅)
 
Theoremdfatafv2eqfv 47210 If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
 
Theoremafv2rnfveq 47211 If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹𝐴))
 
Theoremafv20fv0 47212 If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) = ∅ → (𝐹𝐴) = ∅)
 
Theoremafv2fvn0fveq 47213 If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022.)
((𝐹𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹𝐴))
 
Theoremafv2fv0 47214 If the function's value at an argument is the empty set, then the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
((𝐹𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
 
Theoremafv2fv0b 47215 The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.)
((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
 
Theoremafv2fv0xorb 47216 If a set is in the range of a function, the function's value at an argument is the empty set if and only if the alternate function value at this argument is either the empty set or undefined. (Contributed by AV, 11-Sep-2022.)
(∅ ∈ ran 𝐹 → ((𝐹𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
 
21.48.6  General auxiliary theorems (2)
 
21.48.6.1  Logical conjunction - extension
 
Theoreman4com24 47217 Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜃) ∧ (𝜒𝜓)))
 
21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension
 
Theorem3an4ancom24 47218 Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜃𝜒) ∧ 𝜓))
 
Theorem4an21 47219 Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
(((𝜑𝜓) ∧ 𝜒𝜃) ↔ (𝜓 ∧ (𝜑𝜒𝜃)))
 
21.48.6.3  Negated membership (alternative)
 
Syntaxcnelbr 47220 Extend wff notation to include the 'not element of' relation.
class _∉
 
Definitiondf-nelbr 47221* Define negated membership as binary relation. Analogous to df-eprel 5588 (the membership relation). (Contributed by AV, 26-Dec-2021.)
_∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥𝑦}
 
Theoremdfnelbr2 47222 Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)
_∉ = ((V × V) ∖ E )
 
Theoremnelbr 47223 The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
 
Theoremnelbrim 47224 If a set is related to another set by the negated membership relation, then it is not a member of the other set. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.)
(𝐴 _∉ 𝐵 → ¬ 𝐴𝐵)
 
Theoremnelbrnel 47225 A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))
 
Theoremnelbrnelim 47226 If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
(𝐴 _∉ 𝐵𝐴𝐵)
 
21.48.6.4  The empty set - extension
 
Theoremralralimp 47227* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
 
21.48.6.5  Indexed union and intersection - extension
 
TheoremotiunsndisjX 47228* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
(𝐵𝑋Disj 𝑎𝑉 𝑐𝑊 {⟨𝑎, 𝐵, 𝑐⟩})
 
21.48.6.6  Functions - extension
 
Theoremfvifeq 47229 Equality of function values with conditional arguments, see also fvif 6922. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))
 
Theoremrnfdmpr 47230 The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹𝑋), (𝐹𝑌)}))
 
Theoremimarnf1pr 47231 The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋𝑉𝑌𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸𝐸:dom 𝐸𝑅) ∧ ((𝐸‘(𝐹𝑋)) = 𝐴 ∧ (𝐸‘(𝐹𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵}))
 
Theoremfunop1 47232* A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.)
(∃𝑥𝑦 𝐹 = ⟨𝑥, 𝑦⟩ → (Fun 𝐹 ↔ ∃𝑥𝑦 𝐹 = {⟨𝑥, 𝑦⟩}))
 
Theoremfun2dmnopgexmpl 47233 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Avoid depending on this detail.)
(𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} → ¬ 𝐺 ∈ (V × V))
 
Theoremopabresex0d 47234* A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
 
Theoremopabbrfex0d 47235* A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝜃)    &   ((𝜑𝑥𝐶) → {𝑦𝜃} ∈ 𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
 
Theoremopabresexd 47236* A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} ∈ V)
 
Theoremopabbrfexd 47237* A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
((𝜑𝑥𝑅𝑦) → 𝑥𝐶)    &   ((𝜑𝑥𝑅𝑦) → 𝑦:𝐴𝐵)    &   ((𝜑𝑥𝐶) → 𝐴𝑈)    &   ((𝜑𝑥𝐶) → 𝐵𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V)
 
Theoremf1oresf1orab 47238* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.)
𝐹 = (𝑥𝐴𝐶)    &   (𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐷𝐴)    &   ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝑥𝐷))       (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 
Theoremf1oresf1o 47239* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐷𝐴)    &   (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))       (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 
Theoremf1oresf1o2 47240* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐷𝐴)    &   ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))       (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 
21.48.6.7  Maps-to notation - extension
 
Theoremfvmptrab 47241* Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 7047, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})    &   (𝑥 = 𝑋 → (𝜑𝜓))    &   (𝑥 = 𝑋𝑀 = 𝑁)    &   (𝑋𝑉𝑁 ∈ V)    &   (𝑋𝑉𝑁 = ∅)       (𝐹𝑋) = {𝑦𝑁𝜓}
 
Theoremfvmptrabdm 47242* Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 7047. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.)
𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})    &   (𝑥 = 𝑋 → (𝜑𝜓))    &   (𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)       (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}
 
21.48.6.8  Subtraction - extension
 
Theoremcnambpcma 47243 ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) − 𝐴) = (𝐶𝐵))
 
Theoremcnapbmcpd 47244 ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) − 𝐶) + 𝐷) = (((𝐴 + 𝐷) + 𝐵) − 𝐶))
 
Theoremaddsubeq0 47245 The sum of two complex numbers is equal to the difference of these two complex numbers iff the subtrahend is 0. (Contributed by AV, 8-May-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴𝐵) ↔ 𝐵 = 0))
 
21.48.6.9  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 47246 Addition and subtraction on one side of "less than or equal to". (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) ≤ 𝐴))
 
Theorem2leaddle2 47247 If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶𝐵 < 𝐶) → (𝐴 + 𝐵) < (2 · 𝐶)))
 
Theoremltnltne 47248 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐵 = 𝐴)))
 
Theoremp1lep2 47249 A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ ℝ → (𝑁 + 1) ≤ (𝑁 + 2))
 
Theoremltsubsubaddltsub 47250 If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝐽 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → (𝐽 < ((𝐿𝑀) − 𝑁) ↔ (𝐽 + 𝑀) < (𝐿𝑁)))
 
Theoremzm1nn 47251 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝑁 ∈ ℕ0𝐿 ∈ ℤ) → ((𝐽 ∈ ℝ ∧ 0 ≤ 𝐽𝐽 < ((𝐿𝑁) − 1)) → (𝐿 − 1) ∈ ℕ))
 
21.48.6.10  Imaginary and complex number properties - extension
 
Theoremreaddcnnred 47252 The sum of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))       (𝜑 → (𝐴 + 𝐵) ∉ ℝ)
 
Theoremresubcnnred 47253 The difference of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))       (𝜑 → (𝐴𝐵) ∉ ℝ)
 
Theoremrecnmulnred 47254 The product of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐴 · 𝐵) ∉ ℝ)
 
Theoremcndivrenred 47255 The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ (ℂ ∖ ℝ))    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐵 / 𝐴) ∉ ℝ)
 
Theoremsqrtnegnre 47256 The square root of a negative number is not a real number. (Contributed by AV, 28-Feb-2023.)
((𝑋 ∈ ℝ ∧ 𝑋 < 0) → (√‘𝑋) ∉ ℝ)
 
21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension
 
Theoremnn0resubcl 47257 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ ℝ)
 
21.48.6.12  Integers (as a subset of complex numbers) - extension
 
Theoremzgeltp1eq 47258 If an integer is between another integer and its successor, the integer is equal to the other integer. (Contributed by AV, 30-May-2020.)
((𝐼 ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴𝐼𝐼 < (𝐴 + 1)) → 𝐼 = 𝐴))
 
21.48.6.13  Decimal arithmetic - extension
 
Theorem1t10e1p1e11 47259 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
11 = ((1 · (10↑1)) + 1)
 
Theoremdeccarry 47260 Add 1 to a 2 digit number with carry. This is a special case of decsucc 12771, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g., by applying this theorem three times we get (999 + 1) = 1000. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.) (Proof shortened by AV, 10-Sep-2021.)
(𝐴 ∈ ℕ → (𝐴9 + 1) = (𝐴 + 1)0)
 
21.48.6.14  Upper sets of integers - extension
 
Theoremeluzge0nn0 47261 If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.)
(𝑁 ∈ (ℤ𝑀) → (0 ≤ 𝑀𝑁 ∈ ℕ0))
 
21.48.6.15  Infinity and the extended real number system (cont.) - extension
 
Theoremnltle2tri 47262 Negated extended trichotomy law for 'less than' and 'less than or equal to'. (Contributed by AV, 18-Jul-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵𝐶𝐶𝐴))
 
21.48.6.16  Finite intervals of integers - extension
 
Theoremssfz12 47263 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ ∧ 𝐾𝐿) → ((𝐾...𝐿) ⊆ (𝑀...𝑁) → (𝑀𝐾𝐿𝑁)))
 
Theoremelfz2z 47264 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ (0 ≤ 𝐾𝐾𝑁)))
 
Theorem2elfz3nn0 47265 If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0))
 
Theoremfz0addcom 47266 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theorem2elfz2melfz 47267 If the sum of two integers of a 0-based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0-based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁𝐴)) ∈ (0...𝐴)))
 
Theoremfz0addge0 47268 The sum of two integers in 0-based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ (0...𝑀) ∧ 𝐵 ∈ (0...𝑁)) → 0 ≤ (𝐴 + 𝐵))
 
Theoremelfzlble 47269 Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.)
((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ((𝑁𝑀)...𝑁))
 
Theoremelfzelfzlble 47270 Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑁 < (𝑀 + 𝐾)) → 𝐾 ∈ ((𝑁𝑀)...𝑁))
 
21.48.6.17  Half-open integer ranges - extension
 
Theoremfzopred 47271 Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 13790. (Contributed by AV, 14-Jul-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → (𝑀..^𝑁) = ({𝑀} ∪ ((𝑀 + 1)..^𝑁)))
 
Theoremfzopredsuc 47272 Join a predecessor and a successor to the beginning and the end of an open integer interval. This theorem holds even if 𝑁 = 𝑀 (then (𝑀...𝑁) = {𝑀} = ({𝑀} ∪ ∅) ∪ {𝑀}). (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = (({𝑀} ∪ ((𝑀 + 1)..^𝑁)) ∪ {𝑁}))
 
Theorem1fzopredsuc 47273 Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ ℕ0 → (0...𝑁) = (({0} ∪ (1..^𝑁)) ∪ {𝑁}))
 
Theoremel1fzopredsuc 47274 An element of an open integer interval starting at 1 joined by 0 and a successor at the beginning and the end is either 0 or an element of the open integer interval or the successor. (Contributed by AV, 14-Jul-2020.)
(𝑁 ∈ ℕ0 → (𝐼 ∈ (0...𝑁) ↔ (𝐼 = 0 ∨ 𝐼 ∈ (1..^𝑁) ∨ 𝐼 = 𝑁)))
 
Theoremsubsubelfzo0 47275 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
((𝐴 ∈ (0..^𝑁) ∧ 𝐼 ∈ (0..^𝑁) ∧ ¬ 𝐼 < (𝑁𝐴)) → (𝐼 − (𝑁𝐴)) ∈ (0..^𝐴))
 
Theorem2ffzoeq 47276* Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
(((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
 
21.48.6.18  The modulo (remainder) operation - extension
 
Theoremfldivmod 47277 Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) = ((𝐴 − (𝐴 mod 𝐵)) / 𝐵))
 
Theoremceildivmod 47278 Expressing the ceiling of a division by the modulo operator. (Contributed by AV, 7-Sep-2025.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌈‘(𝐴 / 𝐵)) = ((𝐴 + ((𝐵𝐴) mod 𝐵)) / 𝐵))
 
Theoremceil5half3 47279 The ceiling of half of 5 is 3. (Contributed by AV, 7-Sep-2025.)
(⌈‘(5 / 2)) = 3
 
Theoremsubmodaddmod 47280 Subtraction and addition modulo a positive integer. (Contributed by AV, 7-Sep-2025.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (((𝐴 + 𝐵) mod 𝑁) = ((𝐴𝐶) mod 𝑁) ↔ ((𝐴 + (𝐵 + 𝐶)) mod 𝑁) = (𝐴 mod 𝑁)))
 
Theoremdifltmodne 47281 Two nonnegative integers are not equal modulo a positive modulus if their difference is greater than 0 and less then the modulus. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (1 ≤ (𝐴𝐵) ∧ (𝐴𝐵) < 𝑁)) → (𝐴 mod 𝑁) ≠ (𝐵 mod 𝑁))
 
Theoremzplusmodne 47282 A nonnegative integer is not itself plus a positive integer modulo an integer greater than 1 and the positive integer. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐴 + 𝐾) mod 𝑁) ≠ (𝐴 mod 𝑁))
 
Theoremaddmodne 47283 The sum of a nonnegative integer and a positive integer modulo a number greater than both integers is not equal to the nonnegative integer. (Contributed by AV, 27-Aug-2025.) (Proof shortened by AV, 6-Sep-2025.)
((𝑀 ∈ ℕ ∧ (𝐴 ∈ ℕ0𝐴 < 𝑀) ∧ (𝐵 ∈ ℕ ∧ 𝐵 < 𝑀)) → ((𝐴 + 𝐵) mod 𝑀) ≠ 𝐴)
 
Theoremplusmod5ne 47284 A nonnegative integer is not itself plus a positive integer less than 5 modulo 5. (Contributed by AV, 6-Sep-2025.)
((𝐴 ∈ (0..^5) ∧ 𝐾 ∈ (1..^5)) → ((𝐴 + 𝐾) mod 5) ≠ 𝐴)
 
Theoremzp1modne 47285 An integer is not itself plus 1 modulo an integer greater than 1. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℤ) → ((𝐴 + 1) mod 𝑁) ≠ (𝐴 mod 𝑁))
 
Theoremp1modne 47286 A nonnegative integer is not itself plus 1 modulo an integer greater than 1 and the nonnegative integer. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ (ℤ‘2) ∧ 𝐴 ∈ (0..^𝑁)) → ((𝐴 + 1) mod 𝑁) ≠ 𝐴)
 
Theoremm1modne 47287 A nonnegative integer is not itself minus 1 modulo an integer greater than 1 and the nonnegative integer. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ (ℤ‘2) ∧ 𝐴 ∈ (0..^𝑁)) → ((𝐴 − 1) mod 𝑁) ≠ 𝐴)
 
Theoremminusmod5ne 47288 A nonnegative integer is not itself minus a positive integer less than 5 modulo 5. (Contributed by AV, 7-Sep-2025.)
((𝐴 ∈ (0..^5) ∧ 𝐾 ∈ (1..^5)) → ((𝐴𝐾) mod 5) ≠ 𝐴)
 
Theoremsubmodlt 47289 The difference of an element of a half-open range of nonnegative integers and the upper bound of this range modulo an integer greater than the upper bound. (Contributed by AV, 1-Sep-2025.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ (0..^𝐵) ∧ 𝐵 < 𝑁) → ((𝐴𝐵) mod 𝑁) = ((𝑁 + 𝐴) − 𝐵))
 
Theoremsubmodneaddmod 47290 An integer minus 𝐵 is not itself plus 𝐶 modulo an integer greater than the sum of 𝐵 and 𝐶. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (1 ≤ (𝐵 + 𝐶) ∧ (𝐵 + 𝐶) < 𝑁)) → ((𝐴 + 𝐵) mod 𝑁) ≠ ((𝐴𝐶) mod 𝑁))
 
Theoremm1modnep2mod 47291 A nonnegative integer minus 1 is not itself plus 2 modulo an integer greater than 3 and the nonnegative integer. (Contributed by AV, 6-Sep-2025.)
((𝑁 ∈ (ℤ‘4) ∧ 𝐴 ∈ ℤ) → ((𝐴 − 1) mod 𝑁) ≠ ((𝐴 + 2) mod 𝑁))
 
Theoremminusmodnep2tmod 47292 A nonnegative integer minus a positive integer 1 or 2 is not itself plus 2 times the positive integer modulo 5. (Contributed by AV, 8-Sep-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ (1..^3)) → ((𝐴𝐵) mod 5) ≠ ((𝐴 + (2 · 𝐵)) mod 5))
 
Theoremm1mod0mod1 47293 An integer decreased by 1 is 0 modulo a positive integer iff the integer is 1 modulo the same modulus. (Contributed by AV, 6-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 < 𝑁) → (((𝐴 − 1) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = 1))
 
Theoremelmod2 47294 An integer modulo 2 is either 0 or 1. (Contributed by AV, 24-May-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
(𝑁 ∈ ℤ → (𝑁 mod 2) ∈ {0, 1})
 
21.48.6.19  The infinite sequence builder "seq"
 
Theoremsmonoord 47295* Ordering relation for a strictly monotonic sequence, increasing case. Analogous to monoord 14069 (except that the case 𝑀 = 𝑁 must be excluded). Duplicate of monoords 45247? (Contributed by AV, 12-Jul-2020.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) < (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑀) < (𝐹𝑁))
 
21.48.6.20  Finite and infinite sums - extension
 
Theoremfsummsndifre 47296* A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 ∈ ℝ)
 
Theoremfsumsplitsndif 47297* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ 𝑋𝐴 ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝑋})𝐵 + 𝑋 / 𝑘𝐵))
 
Theoremfsummmodsndifre 47298* A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∖ {𝑋})(𝐵 mod 𝑁) ∈ ℝ)
 
Theoremfsummmodsnunz 47299* A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) ∈ ℤ)
 
21.48.6.21  Extensible structures - extension
 
Theoremsetsidel 47300 The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.)
(𝜑𝑆𝑉)    &   (𝜑𝐵𝑊)    &   𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49035
  Copyright terms: Public domain < Previous  Next >