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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcytpval 41401* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑇 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))    &   π‘‚ = (odβ€˜π‘‡)    &   π‘ƒ = (Poly1β€˜β„‚fld)    &   π‘‹ = (var1β€˜β„‚fld)    &   π‘„ = (mulGrpβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    β‡’   (𝑁 ∈ β„• β†’ (CytPβ€˜π‘) = (𝑄 Ξ£g (π‘Ÿ ∈ (◑𝑂 β€œ {𝑁}) ↦ (𝑋 βˆ’ (π΄β€˜π‘Ÿ)))))
 
21.29.49  Miscellaneous topology
 
Theoremfgraphopab 41402* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹:𝐴⟢𝐡 β†’ 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡) ∧ (πΉβ€˜π‘Ž) = 𝑏)})
 
Theoremfgraphxp 41403* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹:𝐴⟢𝐡 β†’ 𝐹 = {π‘₯ ∈ (𝐴 Γ— 𝐡) ∣ (πΉβ€˜(1st β€˜π‘₯)) = (2nd β€˜π‘₯)})
 
Theoremhausgraph 41404 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹 ∈ (Clsdβ€˜(𝐽 Γ—t 𝐾)))
 
Syntaxctopsep 41405 The class of separable topologies.
class TopSep
 
Syntaxctoplnd 41406 The class of LindelΓΆf topologies.
class TopLnd
 
Definitiondf-topsep 41407* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopSep = {𝑗 ∈ Top ∣ βˆƒπ‘₯ ∈ 𝒫 βˆͺ 𝑗(π‘₯ β‰Ό Ο‰ ∧ ((clsβ€˜π‘—)β€˜π‘₯) = βˆͺ 𝑗)}
 
Definitiondf-toplnd 41408* A topology is LindelΓΆf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopLnd = {π‘₯ ∈ Top ∣ βˆ€π‘¦ ∈ 𝒫 π‘₯(βˆͺ π‘₯ = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ 𝒫 π‘₯(𝑧 β‰Ό Ο‰ ∧ βˆͺ π‘₯ = βˆͺ 𝑧))}
 
21.30  Mathbox for Noam Pasman
 
Theoremr1sssucd 41409 Deductive form of r1sssuc 9652. (Contributed by Noam Pasman, 19-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ On)    β‡’   (πœ‘ β†’ (𝑅1β€˜π΄) βŠ† (𝑅1β€˜suc 𝐴))
 
21.31  Mathbox for Jon Pennant
 
Theoremiocunico 41410 Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (𝐴 < 𝐡 ∧ 𝐡 < 𝐢)) β†’ ((𝐴(,]𝐡) βˆͺ (𝐡[,)𝐢)) = (𝐴(,)𝐢))
 
Theoremiocinico 41411 The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (𝐴 < 𝐡 ∧ 𝐡 < 𝐢)) β†’ ((𝐴(,]𝐡) ∩ (𝐡[,)𝐢)) = {𝐡})
 
Theoremiocmbl 41412 An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ) β†’ (𝐴(,]𝐡) ∈ dom vol)
 
Theoremcnioobibld 41413* A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider 𝐹 = (π‘₯ ∈ (0(,)1) ↦ (1 / π‘₯)). See cniccibl 25127 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ dom 𝐹(absβ€˜(πΉβ€˜π‘¦)) ≀ π‘₯)    β‡’   (πœ‘ β†’ 𝐹 ∈ 𝐿1)
 
Theoremarearect 41414 The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ Γ— ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    &   πΆ ∈ ℝ    &   π· ∈ ℝ    &   π΄ ≀ 𝐡    &   πΆ ≀ 𝐷    &   π‘† = ((𝐴[,]𝐡) Γ— (𝐢[,]𝐷))    β‡’   (areaβ€˜π‘†) = ((𝐡 βˆ’ 𝐴) Β· (𝐷 βˆ’ 𝐢))
 
Theoremareaquad 41415* The area of a quadrilateral with two sides which are parallel to the y-axis in (ℝ Γ— ℝ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    &   πΆ ∈ ℝ    &   π· ∈ ℝ    &   πΈ ∈ ℝ    &   πΉ ∈ ℝ    &   π΄ < 𝐡    &   πΆ ≀ 𝐸    &   π· ≀ 𝐹    &   π‘ˆ = (𝐢 + (((π‘₯ βˆ’ 𝐴) / (𝐡 βˆ’ 𝐴)) Β· (𝐷 βˆ’ 𝐢)))    &   π‘‰ = (𝐸 + (((π‘₯ βˆ’ 𝐴) / (𝐡 βˆ’ 𝐴)) Β· (𝐹 βˆ’ 𝐸)))    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴[,]𝐡) ∧ 𝑦 ∈ (π‘ˆ[,]𝑉))}    β‡’   (areaβ€˜π‘†) = ((((𝐹 + 𝐸) / 2) βˆ’ ((𝐷 + 𝐢) / 2)) Β· (𝐡 βˆ’ 𝐴))
 
21.32  Mathbox for Richard Penner
 
21.32.1  Natural addition of Cantor normal forms
 
Theoremomlimcl2 41416 The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.)
(((𝐴 ∈ On ∧ (𝐡 ∈ 𝐢 ∧ Lim 𝐡)) ∧ βˆ… ∈ 𝐴) β†’ Lim (𝐡 Β·o 𝐴))
 
Theoremoawordex2 41417* If 𝐢 is between 𝐴 (inclusive) and (𝐴 +o 𝐡) (exclusive), there is an ordinal which equals 𝐢 when summed to 𝐴. This is a slightly different statement than oawordex 8471 or oawordeu 8469. (Contributed by RP, 7-Jan-2025.)
(((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (𝐴 βŠ† 𝐢 ∧ 𝐢 ∈ (𝐴 +o 𝐡))) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝐴 +o π‘₯) = 𝐢)
 
Theoremnnawordexg 41418* If an ordinal, 𝐡, is in a half-open interval between some 𝐴 and the next limit ordinal, 𝐡 is the sum of the 𝐴 and some natural number. This weakens the antecedent of nnawordex 8551. (Contributed by RP, 7-Jan-2025.)
((𝐴 ∈ On ∧ 𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ (𝐴 +o Ο‰)) β†’ βˆƒπ‘₯ ∈ Ο‰ (𝐴 +o π‘₯) = 𝐡)
 
Theoremsucclg 41419 Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.)
((𝐴 ∈ 𝐡 ∧ (𝐡 = βˆ… ∨ (𝐡 = (Ο‰ Β·o 𝐢) ∧ 𝐢 ∈ (On βˆ– 1o)))) β†’ suc 𝐴 ∈ 𝐡)
 
Theoremdflim5 41420* A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)
(Lim 𝐴 ↔ (𝐴 = On ∨ βˆƒπ‘₯ ∈ (On βˆ– 1o)𝐴 = (Ο‰ Β·o π‘₯)))
 
Theoremoacl2g 41421 Closure law for ordinal addition. Here we show that ordinal addition is closed within the empty set or any ordinal power of omega. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐢) ∧ (𝐢 = βˆ… ∨ (𝐢 = (Ο‰ ↑o 𝐷) ∧ 𝐷 ∈ On))) β†’ (𝐴 +o 𝐡) ∈ 𝐢)
 
Theoremomabs2 41422 Ordinal multiplication by a larger ordinal is absorbed when the larger ordinal is either 2 or Ο‰ raised to some power of Ο‰. (Contributed by RP, 12-Jan-2025.)
(((𝐴 ∈ 𝐡 ∧ βˆ… ∈ 𝐴) ∧ (𝐡 = βˆ… ∨ 𝐡 = 2o ∨ (𝐡 = (Ο‰ ↑o (Ο‰ ↑o 𝐢)) ∧ 𝐢 ∈ On))) β†’ (𝐴 Β·o 𝐡) = 𝐡)
 
Theoremomcl2 41423 Closure law for ordinal multiplication. (Contributed by RP, 12-Jan-2025.)
(((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐢) ∧ (𝐢 = βˆ… ∨ (𝐢 = (Ο‰ ↑o (Ο‰ ↑o 𝐷)) ∧ 𝐷 ∈ On))) β†’ (𝐴 Β·o 𝐡) ∈ 𝐢)
 
Theoremomcl3g 41424 Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)
(((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐢) ∧ (𝐢 ∈ 3o ∨ (𝐢 = (Ο‰ ↑o (Ο‰ ↑o 𝐷)) ∧ 𝐷 ∈ On))) β†’ (𝐴 Β·o 𝐡) ∈ 𝐢)
 
Theoremofoafg 41425* Addition operator for functions from sets into ordinals results in a function from the intersection of sets into an ordinal. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š ∧ 𝐢 = (𝐴 ∩ 𝐡)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = βˆͺ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) β†’ ( ∘f +o β†Ύ ((𝐷 ↑m 𝐴) Γ— (𝐸 ↑m 𝐡))):((𝐷 ↑m 𝐴) Γ— (𝐸 ↑m 𝐡))⟢(𝐹 ↑m 𝐢))
 
Theoremofoaf 41426 Addition operator for functions from sets into power of omega results in a function from the intersection of sets to that power of omega. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š ∧ 𝐢 = (𝐴 ∩ 𝐡)) ∧ (𝐷 ∈ On ∧ 𝐸 = (Ο‰ ↑o 𝐷))) β†’ ( ∘f +o β†Ύ ((𝐸 ↑m 𝐴) Γ— (𝐸 ↑m 𝐡))):((𝐸 ↑m 𝐴) Γ— (𝐸 ↑m 𝐡))⟢(𝐸 ↑m 𝐢))
 
Theoremofoafo 41427 Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)
((𝐴 ∈ 𝑉 ∧ (𝐡 ∈ On ∧ 𝐢 = (Ο‰ ↑o 𝐡))) β†’ ( ∘f +o β†Ύ ((𝐢 ↑m 𝐴) Γ— (𝐢 ↑m 𝐴))):((𝐢 ↑m 𝐴) Γ— (𝐢 ↑m 𝐴))–ontoβ†’(𝐢 ↑m 𝐴))
 
Theoremofoacl 41428 Closure law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ (𝐡 ∈ On ∧ 𝐢 = (Ο‰ ↑o 𝐡))) ∧ (𝐹 ∈ (𝐢 ↑m 𝐴) ∧ 𝐺 ∈ (𝐢 ↑m 𝐴))) β†’ (𝐹 ∘f +o 𝐺) ∈ (𝐢 ↑m 𝐴))
 
Theoremofoaid1 41429 Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ (𝐹 ∘f +o (𝐴 Γ— {βˆ…})) = 𝐹)
 
Theoremofoaid2 41430 Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ ((𝐴 Γ— {βˆ…}) ∘f +o 𝐹) = 𝐹)
 
Theoremofoaass 41431 Component-wise addition of ordinal-yielding functions is associative. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ (𝐹 ∈ (𝐡 ↑m 𝐴) ∧ 𝐺 ∈ (𝐡 ↑m 𝐴) ∧ 𝐻 ∈ (𝐡 ↑m 𝐴))) β†’ ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
 
Theoremofoacom 41432 Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (Ο‰ ↑m 𝐴) ∧ 𝐺 ∈ (Ο‰ ↑m 𝐴))) β†’ (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
 
Theoremnaddcnff 41433 Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) β†’ ( ∘f +o β†Ύ (𝑆 Γ— 𝑆)):(𝑆 Γ— 𝑆)βŸΆπ‘†)
 
Theoremnaddcnffn 41434 Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) β†’ ( ∘f +o β†Ύ (𝑆 Γ— 𝑆)) Fn (𝑆 Γ— 𝑆))
 
Theoremnaddcnffo 41435 Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) β†’ ( ∘f +o β†Ύ (𝑆 Γ— 𝑆)):(𝑆 Γ— 𝑆)–onto→𝑆)
 
Theoremnaddcnfcl 41436 Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) β†’ (𝐹 ∘f +o 𝐺) ∈ 𝑆)
 
Theoremnaddcnfcom 41437 Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) β†’ (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
 
Theoremnaddcnfid1 41438 Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) β†’ (𝐹 ∘f +o (𝑋 Γ— {βˆ…})) = 𝐹)
 
Theoremnaddcnfid2 41439 Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) β†’ ((𝑋 Γ— {βˆ…}) ∘f +o 𝐹) = 𝐹)
 
Theoremnaddcnfass 41440 Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) β†’ ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
 
21.32.2  Surreal Contributions
 
Theoremabeqabi 41441 Generalized condition for a class abstraction to be equal to some class. (Contributed by RP, 2-Sep-2024.)
𝐴 = {π‘₯ ∣ πœ“}    β‡’   ({π‘₯ ∣ πœ‘} = 𝐴 ↔ βˆ€π‘₯(πœ‘ ↔ πœ“))
 
Theoremabpr 41442* Condition for a class abstraction to be a pair. (Contributed by RP, 25-Aug-2024.)
({π‘₯ ∣ πœ‘} = {π‘Œ, 𝑍} ↔ βˆ€π‘₯(πœ‘ ↔ (π‘₯ = π‘Œ ∨ π‘₯ = 𝑍)))
 
Theoremabtp 41443* Condition for a class abstraction to be a triple. (Contributed by RP, 25-Aug-2024.)
({π‘₯ ∣ πœ‘} = {𝑋, π‘Œ, 𝑍} ↔ βˆ€π‘₯(πœ‘ ↔ (π‘₯ = 𝑋 ∨ π‘₯ = π‘Œ ∨ π‘₯ = 𝑍)))
 
Theoremralopabb 41444* Restricted universal quantification over an ordered-pair class abstraction. (Contributed by RP, 25-Sep-2024.)
𝑂 = {⟨π‘₯, π‘¦βŸ© ∣ πœ‘}    &   (π‘œ = ⟨π‘₯, π‘¦βŸ© β†’ (πœ“ ↔ πœ’))    β‡’   (βˆ€π‘œ ∈ 𝑂 πœ“ ↔ βˆ€π‘₯βˆ€π‘¦(πœ‘ β†’ πœ’))
 
Theorembropabg 41445* Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27047. (Contributed by RP, 26-Sep-2024.)
(π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝐡 β†’ (πœ“ ↔ πœ’))    &   π‘… = {⟨π‘₯, π‘¦βŸ© ∣ πœ‘}    β‡’   (𝐴𝑅𝐡 ↔ ((𝐴 ∈ V ∧ 𝐡 ∈ V) ∧ πœ’))
 
Theoremfpwfvss 41446 Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024.)
𝐹:πΆβŸΆπ’« 𝐡    β‡’   (πΉβ€˜π΄) βŠ† 𝐡
 
Theoremsdomne0 41447 A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.)
(𝐡 β‰Ί 𝐴 β†’ 𝐴 β‰  βˆ…)
 
Theoremsdomne0d 41448 A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024.)
(πœ‘ β†’ 𝐡 β‰Ί 𝐴)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐴 β‰  βˆ…)
 
Theoremsafesnsupfiss 41449 If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝑅 Or 𝐴)    β‡’   (πœ‘ β†’ if(𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡) βŠ† 𝐡)
 
Theoremsafesnsupfiub 41450* If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝑅 Or 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐢 π‘₯𝑅𝑦)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ if (𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡)βˆ€π‘¦ ∈ 𝐢 π‘₯𝑅𝑦)
 
Theoremsafesnsupfidom1o 41451 If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    β‡’   (πœ‘ β†’ if(𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡) β‰Ό 1o)
 
Theoremsafesnsupfilb 41452* If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 3-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝑅 Or 𝐴)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐡 βˆ– if(𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡))βˆ€π‘¦ ∈ if (𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡)π‘₯𝑅𝑦)
 
Theoremisoeq145d 41453 Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐢)    &   (πœ‘ β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐢, 𝐷)))
 
Theoremresisoeq45d 41454 Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)
(πœ‘ β†’ 𝐴 = 𝐢)    &   (πœ‘ β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ ((𝐹 β†Ύ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐡) ↔ (𝐹 β†Ύ 𝐢) Isom 𝑅, 𝑆 (𝐢, 𝐷)))
 
Theoremnegslem1 41455 An equivalence between identically restricted order-reversing self-isometries. (Contributed by RP, 30-Sep-2024.)
(𝐴 = 𝐡 β†’ ((𝐹 β†Ύ 𝐴) Isom 𝑅, ◑𝑅(𝐴, 𝐴) ↔ (𝐹 β†Ύ 𝐡) Isom 𝑅, ◑𝑅(𝐡, 𝐡)))
 
Theoremnvocnvb 41456* Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.)
((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (πΉβ€˜(πΉβ€˜π‘₯)) = π‘₯))
 
Theoremrp-brsslt 41457* Binary relation form of a relation, <, which has been extended from relation 𝑅 to subsets of class 𝑆. Usually, we will assume 𝑅 Or 𝑆. Definition in [Alling], p. 2. Generalization of brsslt 27047. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    β‡’   (𝐴 < 𝐡 ↔ ((𝐴 ∈ V ∧ 𝐡 ∈ V) ∧ (𝐴 βŠ† 𝑆 ∧ 𝐡 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 π‘₯𝑅𝑦)))
 
Theoremnla0002 41458* Extending a linear order to subsets, the empty set is less than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    &   (πœ‘ β†’ 𝐴 ∈ V)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    β‡’   (πœ‘ β†’ βˆ… < 𝐴)
 
Theoremnla0003 41459* Extending a linear order to subsets, the empty set is greater than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    &   (πœ‘ β†’ 𝐴 ∈ V)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    β‡’   (πœ‘ β†’ 𝐴 < βˆ…)
 
Theoremnla0001 41460* Extending a linear order to subsets, the empty set is less than itself. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    β‡’   (πœ‘ β†’ βˆ… < βˆ…)
 
Theoremfaosnf0.11b 41461* 𝐡 is called a non-limit ordinal if it is not a limit ordinal. (Contributed by RP, 27-Sep-2023.)

Alling, Norman L. "Fundamentals of Analysis Over Surreal Numbers Fields." The Rocky Mountain Journal of Mathematics 19, no. 3 (1989): 565-73. http://www.jstor.org/stable/44237243.

((Ord 𝐴 ∧ Β¬ Lim 𝐴 ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯)
 
Theoremdfno2 41462 A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.)
No = {𝑓 ∈ 𝒫 (On Γ— {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
 
Theoremonnog 41463 Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
((𝐴 ∈ On ∧ 𝐡 ∈ {1o, 2o}) β†’ (𝐴 Γ— {𝐡}) ∈ No )
 
Theoremonnobdayg 41464 Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.)
((𝐴 ∈ On ∧ 𝐡 ∈ {1o, 2o}) β†’ ( bday β€˜(𝐴 Γ— {𝐡})) = 𝐴)
 
Theorembdaybndex 41465 Bounds formed from the birthday are surreal numbers. (Contributed by RP, 21-Sep-2023.)
((𝐴 ∈ No ∧ 𝐡 = ( bday β€˜π΄) ∧ 𝐢 ∈ {1o, 2o}) β†’ (𝐡 Γ— {𝐢}) ∈ No )
 
Theorembdaybndbday 41466 Bounds formed from the birthday have the same birthday. (Contributed by RP, 30-Sep-2023.)
((𝐴 ∈ No ∧ 𝐡 = ( bday β€˜π΄) ∧ 𝐢 ∈ {1o, 2o}) β†’ ( bday β€˜(𝐡 Γ— {𝐢})) = ( bday β€˜π΄))
 
Theoremonno 41467 Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(𝐴 ∈ On β†’ (𝐴 Γ— {2o}) ∈ No )
 
Theoremonnoi 41468 Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
𝐴 ∈ On    β‡’   (𝐴 Γ— {2o}) ∈ No
 
Theorem0no 41469 Ordinal zero maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
βˆ… ∈ No
 
Theorem1no 41470 Ordinal one maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(1o Γ— {2o}) ∈ No
 
Theorem2no 41471 Ordinal two maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(2o Γ— {2o}) ∈ No
 
Theorem3no 41472 Ordinal three maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(3o Γ— {2o}) ∈ No
 
Theorem4no 41473 Ordinal four maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(4o Γ— {2o}) ∈ No
 
Theoremfnimafnex 41474 The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.)
𝐹 Fn 𝐡    β‡’   (𝐹 β€œ (πΊβ€˜π΄)) ∈ V
 
21.32.3  Short Studies
 
Theoremnlimsuc 41475 A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
(𝐴 ∈ On β†’ Β¬ Lim suc 𝐴)
 
Theoremnlim1NEW 41476 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
Β¬ Lim 1o
 
Theoremnlim2NEW 41477 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
Β¬ Lim 2o
 
Theoremnlim3 41478 3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
Β¬ Lim 3o
 
Theoremnlim4 41479 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
Β¬ Lim 4o
 
Theoremoa1un 41480 Given 𝐴 ∈ On, let 𝐴 +o 1o be defined to be the union of 𝐴 and {𝐴}. Compare with oa1suc 8444. (Contributed by RP, 27-Sep-2023.)
(𝐴 ∈ On β†’ (𝐴 +o 1o) = (𝐴 βˆͺ {𝐴}))
 
Theoremoa1cl 41481 𝐴 +o 1o is in On. (Contributed by RP, 27-Sep-2023.)
(𝐴 ∈ On β†’ (𝐴 +o 1o) ∈ On)
 
Theorem0finon 41482 0 is a finite ordinal. See peano1 7815. (Contributed by RP, 27-Sep-2023.)
βˆ… ∈ (On ∩ Fin)
 
Theorem1finon 41483 1 is a finite ordinal. See 1onn 8553. (Contributed by RP, 27-Sep-2023.)
1o ∈ (On ∩ Fin)
 
Theorem2finon 41484 2 is a finite ordinal. See 1onn 8553. (Contributed by RP, 27-Sep-2023.)
2o ∈ (On ∩ Fin)
 
Theorem3finon 41485 3 is a finite ordinal. See 1onn 8553. (Contributed by RP, 27-Sep-2023.)
3o ∈ (On ∩ Fin)
 
Theorem4finon 41486 4 is a finite ordinal. See 1onn 8553. (Contributed by RP, 27-Sep-2023.)
4o ∈ (On ∩ Fin)
 
Theoremfinona1cl 41487 The finite ordinals are closed under the add one operation. (Contributed by RP, 27-Sep-2023.)
(𝑁 ∈ (On ∩ Fin) β†’ (𝑁 +o 1o) ∈ (On ∩ Fin))
 
Theoremfinonex 41488 The finite ordinals are a set. See also onprc 7702 and fiprc 8922 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.)
(On ∩ Fin) ∈ V
 
Theoremfzunt 41489 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.)
(((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐾 ≀ 𝑀 ∧ 𝑀 ≀ 𝑁)) β†’ ((𝐾...𝑀) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
Theoremfzuntd 41490 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    &   (πœ‘ β†’ 𝑀 ≀ 𝑁)    β‡’   (πœ‘ β†’ ((𝐾...𝑀) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
Theoremfzunt1d 41491 Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    &   (πœ‘ β†’ 𝑀 ≀ 𝐿)    &   (πœ‘ β†’ 𝐿 ≀ 𝑁)    β‡’   (πœ‘ β†’ ((𝐾...𝐿) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
Theoremfzuntgd 41492 Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    &   (πœ‘ β†’ 𝑀 ≀ (𝐿 + 1))    &   (πœ‘ β†’ 𝐿 ≀ 𝑁)    β‡’   (πœ‘ β†’ ((𝐾...𝐿) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
21.32.3.1  Additional work on conditional logical operator
 
Theoremifpan123g 41493 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(πœ‘, πœ’, 𝜏) ∧ if-(πœ“, πœƒ, πœ‚)) ↔ (((Β¬ πœ‘ ∨ πœ’) ∧ (πœ‘ ∨ 𝜏)) ∧ ((Β¬ πœ“ ∨ πœƒ) ∧ (πœ“ ∨ πœ‚))))
 
Theoremifpan23 41494 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
((if-(πœ‘, πœ“, πœ’) ∧ if-(πœ‘, πœƒ, 𝜏)) ↔ if-(πœ‘, (πœ“ ∧ πœƒ), (πœ’ ∧ 𝜏)))
 
Theoremifpdfor2 41495 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ∨ πœ“) ↔ if-(πœ‘, πœ‘, πœ“))
 
Theoremifporcor 41496 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
(if-(πœ‘, πœ‘, πœ“) ↔ if-(πœ“, πœ“, πœ‘))
 
Theoremifpdfan2 41497 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((πœ‘ ∧ πœ“) ↔ if-(πœ‘, πœ“, πœ‘))
 
Theoremifpancor 41498 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(πœ‘, πœ“, πœ‘) ↔ if-(πœ“, πœ‘, πœ“))
 
Theoremifpdfor 41499 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ∨ πœ“) ↔ if-(πœ‘, ⊀, πœ“))
 
Theoremifpdfan 41500 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ∧ πœ“) ↔ if-(πœ‘, πœ“, βŠ₯))
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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 46901-46997
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