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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | abvpropd2 31601 | Weaker version of abvpropd 20224. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
β’ (π β (BaseβπΎ) = (BaseβπΏ)) & β’ (π β (+gβπΎ) = (+gβπΏ)) & β’ (π β (.rβπΎ) = (.rβπΏ)) β β’ (π β (AbsValβπΎ) = (AbsValβπΏ)) | ||
Theorem | oppgle 31602 | less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
β’ π = (oppgβπ ) & β’ β€ = (leβπ ) β β’ β€ = (leβπ) | ||
Theorem | oppgleOLD 31603 | Obsolete version of oppgle 31602 as of 27-Oct-2024. less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (oppgβπ ) & β’ β€ = (leβπ ) β β’ β€ = (leβπ) | ||
Theorem | oppglt 31604 | less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
β’ π = (oppgβπ ) & β’ < = (ltβπ ) β β’ (π β π β < = (ltβπ)) | ||
Theorem | ressprs 31605 | The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
β’ π΅ = (BaseβπΎ) β β’ ((πΎ β Proset β§ π΄ β π΅) β (πΎ βΎs π΄) β Proset ) | ||
Theorem | oduprs 31606 | Being a proset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
β’ π· = (ODualβπΎ) β β’ (πΎ β Proset β π· β Proset ) | ||
Theorem | posrasymb 31607 | A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = ((leβπΎ) β© (π΅ Γ π΅)) β β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) | ||
Theorem | resspos 31608 | The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
β’ ((πΉ β Poset β§ π΄ β π) β (πΉ βΎs π΄) β Poset) | ||
Theorem | resstos 31609 | The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
β’ ((πΉ β Toset β§ π΄ β π) β (πΉ βΎs π΄) β Toset) | ||
Theorem | odutos 31610 | Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
β’ π· = (ODualβπΎ) β β’ (πΎ β Toset β π· β Toset) | ||
Theorem | tlt2 31611 | In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ < = (ltβπΎ) β β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π β€ π β¨ π < π)) | ||
Theorem | tlt3 31612 | In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
β’ π΅ = (BaseβπΎ) & β’ < = (ltβπΎ) β β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π = π β¨ π < π β¨ π < π)) | ||
Theorem | trleile 31613 | In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = ((leβπΎ) β© (π΅ Γ π΅)) β β’ ((πΎ β Toset β§ π β π΅ β§ π β π΅) β (π β€ π β¨ π β€ π)) | ||
Theorem | toslublem 31614* | Lemma for toslub 31615 and xrsclat 31653. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.) |
β’ π΅ = (BaseβπΎ) & β’ < = (ltβπΎ) & β’ (π β πΎ β Toset) & β’ (π β π΄ β π΅) & β’ β€ = (leβπΎ) β β’ ((π β§ π β π΅) β ((βπ β π΄ π β€ π β§ βπ β π΅ (βπ β π΄ π β€ π β π β€ π)) β (βπ β π΄ Β¬ π < π β§ βπ β π΅ (π < π β βπ β π΄ π < π)))) | ||
Theorem | toslub 31615 | In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(π΄, π΅, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.) |
β’ π΅ = (BaseβπΎ) & β’ < = (ltβπΎ) & β’ (π β πΎ β Toset) & β’ (π β π΄ β π΅) β β’ (π β ((lubβπΎ)βπ΄) = sup(π΄, π΅, < )) | ||
Theorem | tosglblem 31616* | Lemma for tosglb 31617 and xrsclat 31653. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.) |
β’ π΅ = (BaseβπΎ) & β’ < = (ltβπΎ) & β’ (π β πΎ β Toset) & β’ (π β π΄ β π΅) & β’ β€ = (leβπΎ) β β’ ((π β§ π β π΅) β ((βπ β π΄ π β€ π β§ βπ β π΅ (βπ β π΄ π β€ π β π β€ π)) β (βπ β π΄ Β¬ πβ‘ < π β§ βπ β π΅ (πβ‘ < π β βπ β π΄ πβ‘ < π)))) | ||
Theorem | tosglb 31617 | Same theorem as toslub 31615, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.) |
β’ π΅ = (BaseβπΎ) & β’ < = (ltβπΎ) & β’ (π β πΎ β Toset) & β’ (π β π΄ β π΅) β β’ (π β ((glbβπΎ)βπ΄) = inf(π΄, π΅, < )) | ||
Theorem | clatp0cl 31618 | The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ 0 = (0.βπ) β β’ (π β CLat β 0 β π΅) | ||
Theorem | clatp1cl 31619 | The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.) |
β’ π΅ = (Baseβπ) & β’ 1 = (1.βπ) β β’ (π β CLat β 1 β π΅) | ||
Syntax | cmnt 31620 | Extend class notation with monotone functions. |
class Monot | ||
Syntax | cmgc 31621 | Extend class notation with the monotone Galois connection. |
class MGalConn | ||
Definition | df-mnt 31622* | Define a monotone function between two ordered sets. (Contributed by Thierry Arnoux, 20-Apr-2024.) |
β’ Monot = (π£ β V, π€ β V β¦ β¦(Baseβπ£) / πβ¦{π β ((Baseβπ€) βm π) β£ βπ₯ β π βπ¦ β π (π₯(leβπ£)π¦ β (πβπ₯)(leβπ€)(πβπ¦))}) | ||
Definition | df-mgc 31623* | Define monotone Galois connections. See mgcval 31629 for an expanded version. (Contributed by Thierry Arnoux, 20-Apr-2024.) |
β’ MGalConn = (π£ β V, π€ β V β¦ β¦(Baseβπ£) / πβ¦β¦(Baseβπ€) / πβ¦{β¨π, πβ© β£ ((π β (π βm π) β§ π β (π βm π)) β§ βπ₯ β π βπ¦ β π ((πβπ₯)(leβπ€)π¦ β π₯(leβπ£)(πβπ¦)))}) | ||
Theorem | mntoval 31624* | Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) β β’ ((π β π β§ π β π) β (πMonotπ) = {π β (π΅ βm π΄) β£ βπ₯ β π΄ βπ¦ β π΄ (π₯ β€ π¦ β (πβπ₯) β² (πβπ¦))}) | ||
Theorem | ismnt 31625* | Express the statement "πΉ is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) β β’ ((π β π β§ π β π) β (πΉ β (πMonotπ) β (πΉ:π΄βΆπ΅ β§ βπ₯ β π΄ βπ¦ β π΄ (π₯ β€ π¦ β (πΉβπ₯) β² (πΉβπ¦))))) | ||
Theorem | ismntd 31626 | Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ (π β π β πΆ) & β’ (π β π β π·) & β’ (π β πΉ β (πMonotπ)) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β π β€ π) β β’ (π β (πΉβπ) β² (πΉβπ)) | ||
Theorem | mntf 31627 | A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) β β’ ((π β π β§ π β π β§ πΉ β (πMonotπ)) β πΉ:π΄βΆπ΅) | ||
Theorem | mgcoval 31628* | Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) β β’ ((π β π β§ π β π) β (πMGalConnπ) = {β¨π, πβ© β£ ((π β (π΅ βm π΄) β§ π β (π΄ βm π΅)) β§ βπ₯ β π΄ βπ¦ β π΅ ((πβπ₯) β² π¦ β π₯ β€ (πβπ¦)))}) | ||
Theorem | mgcval 31629* |
Monotone Galois connection between two functions πΉ and πΊ. If
this relation is satisfied, πΉ is called the lower adjoint of πΊ,
and πΊ is called the upper adjoint of πΉ.
Technically, this is implemented as an operation taking a pair of structures π and π, expected to be posets, which gives a relation between pairs of functions πΉ and πΊ. If such a relation exists, it can be proven to be unique. Galois connections generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields. (Contributed by Thierry Arnoux, 23-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) β β’ (π β (πΉπ»πΊ β ((πΉ:π΄βΆπ΅ β§ πΊ:π΅βΆπ΄) β§ βπ₯ β π΄ βπ¦ β π΅ ((πΉβπ₯) β² π¦ β π₯ β€ (πΊβπ¦))))) | ||
Theorem | mgcf1 31630 | The lower adjoint πΉ of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) β β’ (π β πΉ:π΄βΆπ΅) | ||
Theorem | mgcf2 31631 | The upper adjoint πΊ of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) β β’ (π β πΊ:π΅βΆπ΄) | ||
Theorem | mgccole1 31632 | An inequality for the kernel operator πΊ β πΉ. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) & β’ (π β π β π΄) β β’ (π β π β€ (πΊβ(πΉβπ))) | ||
Theorem | mgccole2 31633 | Inequality for the closure operator (πΉ β πΊ) of the Galois connection π». (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) & β’ (π β π β π΅) β β’ (π β (πΉβ(πΊβπ)) β² π) | ||
Theorem | mgcmnt1 31634 | The lower adjoint πΉ of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β π β€ π) β β’ (π β (πΉβπ) β² (πΉβπ)) | ||
Theorem | mgcmnt2 31635 | The upper adjoint πΊ of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β² π) β β’ (π β (πΊβπ) β€ (πΊβπ)) | ||
Theorem | mgcmntco 31636* | A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) & β’ πΆ = (Baseβπ) & β’ < = (leβπ) & β’ (π β π β Proset ) & β’ (π β πΎ β (πMonotπ)) & β’ (π β πΏ β (πMonotπ)) β β’ (π β (βπ₯ β π΄ (πΎβπ₯) < (πΏβ(πΉβπ₯)) β βπ¦ β π΅ (πΎβ(πΊβπ¦)) < (πΏβπ¦))) | ||
Theorem | dfmgc2lem 31637* | Lemma for dfmgc2, backwards direction. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉ:π΄βΆπ΅) & β’ (π β πΊ:π΅βΆπ΄) & β’ (π β βπ₯ β π΄ βπ¦ β π΄ (π₯ β€ π¦ β (πΉβπ₯) β² (πΉβπ¦))) & β’ (π β βπ’ β π΅ βπ£ β π΅ (π’ β² π£ β (πΊβπ’) β€ (πΊβπ£))) & β’ ((π β§ π₯ β π΄) β π₯ β€ (πΊβ(πΉβπ₯))) & β’ ((π β§ π’ β π΅) β (πΉβ(πΊβπ’)) β² π’) β β’ (π β πΉπ»πΊ) | ||
Theorem | dfmgc2 31638* | Alternate definition of the monotone Galois connection. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) β β’ (π β (πΉπ»πΊ β ((πΉ:π΄βΆπ΅ β§ πΊ:π΅βΆπ΄) β§ ((βπ₯ β π΄ βπ¦ β π΄ (π₯ β€ π¦ β (πΉβπ₯) β² (πΉβπ¦)) β§ βπ’ β π΅ βπ£ β π΅ (π’ β² π£ β (πΊβπ’) β€ (πΊβπ£))) β§ (βπ’ β π΅ (πΉβ(πΊβπ’)) β² π’ β§ βπ₯ β π΄ π₯ β€ (πΊβ(πΉβπ₯))))))) | ||
Theorem | mgcmnt1d 31639 | Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.) |
β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) β β’ (π β πΉ β (πMonotπ)) | ||
Theorem | mgcmnt2d 31640 | Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.) |
β’ π» = (πMGalConnπ) & β’ (π β π β Proset ) & β’ (π β π β Proset ) & β’ (π β πΉπ»πΊ) β β’ (π β πΊ β (πMonotπ)) | ||
Theorem | mgccnv 31641 | The inverse Galois connection is the Galois connection of the dual orders. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ π» = (πMGalConnπ) & β’ π = ((ODualβπ)MGalConn(ODualβπ)) β β’ ((π β Proset β§ π β Proset ) β (πΉπ»πΊ β πΊππΉ)) | ||
Theorem | pwrssmgc 31642* | Given a function πΉ, exhibit a Galois connection between subsets of its domain and subsets of its range. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
β’ πΊ = (π β π« π β¦ (β‘πΉ β π)) & β’ π» = (π β π« π β¦ {π¦ β π β£ (β‘πΉ β {π¦}) β π}) & β’ π = (toIncβπ« π) & β’ π = (toIncβπ« π) & β’ (π β π β π΄) & β’ (π β π β π΅) & β’ (π β πΉ:πβΆπ) β β’ (π β πΊ(πMGalConnπ)π») | ||
Theorem | mgcf1olem1 31643 | Property of a Galois connection, lemma for mgcf1o 31645. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
β’ π» = (πMGalConnπ) & β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ (π β π β Poset) & β’ (π β π β Poset) & β’ (π β πΉπ»πΊ) & β’ (π β π β π΄) β β’ (π β (πΉβ(πΊβ(πΉβπ))) = (πΉβπ)) | ||
Theorem | mgcf1olem2 31644 | Property of a Galois connection, lemma for mgcf1o 31645. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
β’ π» = (πMGalConnπ) & β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ (π β π β Poset) & β’ (π β π β Poset) & β’ (π β πΉπ»πΊ) & β’ (π β π β π΅) β β’ (π β (πΊβ(πΉβ(πΊβπ))) = (πΊβπ)) | ||
Theorem | mgcf1o 31645 | Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
β’ π» = (πMGalConnπ) & β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ β² = (leβπ) & β’ (π β π β Poset) & β’ (π β π β Poset) & β’ (π β πΉπ»πΊ) β β’ (π β (πΉ βΎ ran πΊ) Isom β€ , β² (ran πΊ, ran πΉ)) | ||
Axiom | ax-xrssca 31646 | Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.) |
β’ βfld = (Scalarββ*π ) | ||
Axiom | ax-xrsvsca 31647 | Assume the scalar product of the extended real structure is the extended real number multiplication operation (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.) |
β’ Β·e = ( Β·π ββ*π ) | ||
Theorem | xrs0 31648 | The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13097 and df-xrs 17319), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
β’ 0 = (0gββ*π ) | ||
Theorem | xrslt 31649 | The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ < = (ltββ*π ) | ||
Theorem | xrsinvgval 31650 | The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13097 and df-xrs 17319), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
β’ (π΅ β β* β ((invgββ*π )βπ΅) = -ππ΅) | ||
Theorem | xrsmulgzz 31651 | The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
β’ ((π΄ β β€ β§ π΅ β β*) β (π΄(.gββ*π )π΅) = (π΄ Β·e π΅)) | ||
Theorem | xrstos 31652 | The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.) |
β’ β*π β Toset | ||
Theorem | xrsclat 31653 | The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.) |
β’ β*π β CLat | ||
Theorem | xrsp0 31654 | The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.) |
β’ -β = (0.ββ*π ) | ||
Theorem | xrsp1 31655 | The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) |
β’ +β = (1.ββ*π ) | ||
Theorem | ressmulgnn 31656 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.) |
β’ π» = (πΊ βΎs π΄) & β’ π΄ β (BaseβπΊ) & β’ β = (.gβπΊ) & β’ πΌ = (invgβπΊ) β β’ ((π β β β§ π β π΄) β (π(.gβπ»)π) = (π β π)) | ||
Theorem | ressmulgnn0 31657 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
β’ π» = (πΊ βΎs π΄) & β’ π΄ β (BaseβπΊ) & β’ β = (.gβπΊ) & β’ πΌ = (invgβπΊ) & β’ (0gβπΊ) = (0gβπ») β β’ ((π β β0 β§ π β π΄) β (π(.gβπ»)π) = (π β π)) | ||
Theorem | xrge0base 31658 | The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | ||
Theorem | xrge00 31659 | The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
β’ 0 = (0gβ(β*π βΎs (0[,]+β))) | ||
Theorem | xrge0plusg 31660 | The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.) |
β’ +π = (+gβ(β*π βΎs (0[,]+β))) | ||
Theorem | xrge0le 31661 | The "less than or equal to" relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.) |
β’ β€ = (leβ(β*π βΎs (0[,]+β))) | ||
Theorem | xrge0mulgnn0 31662 | The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
β’ ((π΄ β β0 β§ π΅ β (0[,]+β)) β (π΄(.gβ(β*π βΎs (0[,]+β)))π΅) = (π΄ Β·e π΅)) | ||
Theorem | xrge0addass 31663 | Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.) |
β’ ((π΄ β (0[,]+β) β§ π΅ β (0[,]+β) β§ πΆ β (0[,]+β)) β ((π΄ +π π΅) +π πΆ) = (π΄ +π (π΅ +π πΆ))) | ||
Theorem | xrge0addgt0 31664 | The sum of nonnegative and positive numbers is positive. See addgtge0 11577. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
β’ (((π΄ β (0[,]+β) β§ π΅ β (0[,]+β)) β§ 0 < π΄) β 0 < (π΄ +π π΅)) | ||
Theorem | xrge0adddir 31665 | Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.) |
β’ ((π΄ β (0[,]+β) β§ π΅ β (0[,]+β) β§ πΆ β (0[,]+β)) β ((π΄ +π π΅) Β·e πΆ) = ((π΄ Β·e πΆ) +π (π΅ Β·e πΆ))) | ||
Theorem | xrge0adddi 31666 | Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
β’ ((π΄ β (0[,]+β) β§ π΅ β (0[,]+β) β§ πΆ β (0[,]+β)) β (πΆ Β·e (π΄ +π π΅)) = ((πΆ Β·e π΄) +π (πΆ Β·e π΅))) | ||
Theorem | xrge0npcan 31667 | Extended nonnegative real version of npcan 11344. (Contributed by Thierry Arnoux, 9-Jun-2017.) |
β’ ((π΄ β (0[,]+β) β§ π΅ β (0[,]+β) β§ π΅ β€ π΄) β ((π΄ +π -ππ΅) +π π΅) = π΄) | ||
Theorem | fsumrp0cl 31668* | Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.) |
β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β (0[,)+β)) β β’ (π β Ξ£π β π΄ π΅ β (0[,)+β)) | ||
Theorem | abliso 31669 | The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.) |
β’ ((π β Abel β§ πΉ β (π GrpIso π)) β π β Abel) | ||
Theorem | gsumsubg 31670 | The group sum in a subgroup is the same as the group sum. (Contributed by Thierry Arnoux, 28-May-2023.) |
β’ π» = (πΊ βΎs π΅) & β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆπ΅) & β’ (π β π΅ β (SubGrpβπΊ)) β β’ (π β (πΊ Ξ£g πΉ) = (π» Ξ£g πΉ)) | ||
Theorem | gsumsra 31671 | The group sum in a subring algebra is the same as the ring's group sum. (Contributed by Thierry Arnoux, 28-May-2023.) |
β’ π΄ = ((subringAlg βπ )βπ΅) & β’ (π β πΉ β π) & β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β (Baseβπ )) β β’ (π β (π Ξ£g πΉ) = (π΄ Ξ£g πΉ)) | ||
Theorem | gsummpt2co 31672* | Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ) & β’ (π β π β CMnd) & β’ (π β π΄ β Fin) & β’ (π β πΈ β π) & β’ ((π β§ π₯ β π΄) β πΆ β π΅) & β’ ((π β§ π₯ β π΄) β π· β πΈ) & β’ πΉ = (π₯ β π΄ β¦ π·) β β’ (π β (π Ξ£g (π₯ β π΄ β¦ πΆ)) = (π Ξ£g (π¦ β πΈ β¦ (π Ξ£g (π₯ β (β‘πΉ β {π¦}) β¦ πΆ))))) | ||
Theorem | gsummpt2d 31673* | Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19678. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
β’ β²π§πΆ & β’ β²π¦π & β’ π΅ = (Baseβπ) & β’ (π₯ = β¨π¦, π§β© β πΆ = π·) & β’ (π β Rel π΄) & β’ (π β π΄ β Fin) & β’ (π β π β CMnd) & β’ ((π β§ π₯ β π΄) β πΆ β π΅) β β’ (π β (π Ξ£g (π₯ β π΄ β¦ πΆ)) = (π Ξ£g (π¦ β dom π΄ β¦ (π Ξ£g (π§ β (π΄ β {π¦}) β¦ π·))))) | ||
Theorem | lmodvslmhm 31674* | Scalar multiplication in a left module by a fixed element is a group homomorphism. (Contributed by Thierry Arnoux, 12-Jun-2023.) |
β’ π = (Baseβπ) & β’ πΉ = (Scalarβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (BaseβπΉ) β β’ ((π β LMod β§ π β π) β (π₯ β πΎ β¦ (π₯ Β· π)) β (πΉ GrpHom π)) | ||
Theorem | gsumvsmul1 31675* | Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc1 19953, since every ring is a left module over itself. (Contributed by Thierry Arnoux, 12-Jun-2023.) |
β’ π΅ = (Baseβπ ) & β’ π = (Scalarβπ ) & β’ πΎ = (Baseβπ) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ ) & β’ (π β π β LMod) & β’ (π β π β CMnd) & β’ (π β π΄ β π) & β’ (π β π β π΅) & β’ ((π β§ π β π΄) β π β πΎ) & β’ (π β (π β π΄ β¦ π) finSupp 0 ) β β’ (π β (π Ξ£g (π β π΄ β¦ (π Β· π))) = ((π Ξ£g (π β π΄ β¦ π)) Β· π)) | ||
Theorem | gsummptres 31676* | Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ (π β πΊ β CMnd) & β’ (π β π΄ β Fin) & β’ ((π β§ π₯ β π΄) β πΆ β π΅) & β’ ((π β§ π₯ β (π΄ β π·)) β πΆ = 0 ) β β’ (π β (πΊ Ξ£g (π₯ β π΄ β¦ πΆ)) = (πΊ Ξ£g (π₯ β (π΄ β© π·) β¦ πΆ))) | ||
Theorem | gsummptres2 31677* | Extend a finite group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ (π β πΊ β CMnd) & β’ (π β π΄ β π) & β’ ((π β§ π₯ β (π΄ β π)) β π = 0 ) & β’ (π β π β Fin) & β’ ((π β§ π₯ β π΄) β π β π΅) & β’ (π β π β π΄) β β’ (π β (πΊ Ξ£g (π₯ β π΄ β¦ π)) = (πΊ Ξ£g (π₯ β π β¦ π))) | ||
Theorem | gsumzresunsn 31678 | Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) & β’ π = (CntzβπΊ) & β’ π = (πΉβπ) & β’ (π β πΉ:πΆβΆπ΅) & β’ (π β π΄ β πΆ) & β’ (π β πΊ β Mnd) & β’ (π β π΄ β Fin) & β’ (π β Β¬ π β π΄) & β’ (π β π β πΆ) & β’ (π β π β π΅) & β’ (π β (πΉ β (π΄ βͺ {π})) β (πβ(πΉ β (π΄ βͺ {π})))) β β’ (π β (πΊ Ξ£g (πΉ βΎ (π΄ βͺ {π}))) = ((πΊ Ξ£g (πΉ βΎ π΄)) + π)) | ||
Theorem | gsumpart 31679* | Express a group sum as a double sum, grouping along a (possibly infinite) partition. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ (π β πΊ β CMnd) & β’ (π β π΄ β π) & β’ (π β π β π) & β’ (π β πΉ:π΄βΆπ΅) & β’ (π β πΉ finSupp 0 ) & β’ (π β Disj π₯ β π πΆ) & β’ (π β βͺ π₯ β π πΆ = π΄) β β’ (π β (πΊ Ξ£g πΉ) = (πΊ Ξ£g (π₯ β π β¦ (πΊ Ξ£g (πΉ βΎ πΆ))))) | ||
Theorem | gsumhashmul 31680* | Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ Β· = (.gβπΊ) & β’ (π β πΊ β CMnd) & β’ (π β πΉ:π΄βΆπ΅) & β’ (π β πΉ finSupp 0 ) β β’ (π β (πΊ Ξ£g πΉ) = (πΊ Ξ£g (π₯ β (ran πΉ β { 0 }) β¦ ((β―β(β‘πΉ β {π₯})) Β· π₯)))) | ||
Theorem | xrge0tsmsd 31681* | Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.) |
β’ πΊ = (β*π βΎs (0[,]+β)) & β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆ(0[,]+β)) & β’ (π β π = sup(ran (π β (π« π΄ β© Fin) β¦ (πΊ Ξ£g (πΉ βΎ π ))), β*, < )) β β’ (π β (πΊ tsums πΉ) = {π}) | ||
Theorem | xrge0tsmsbi 31682 | Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.) |
β’ πΊ = (β*π βΎs (0[,]+β)) & β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆ(0[,]+β)) β β’ (π β (πΆ β (πΊ tsums πΉ) β πΆ = βͺ (πΊ tsums πΉ))) | ||
Theorem | xrge0tsmseq 31683 | Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
β’ πΊ = (β*π βΎs (0[,]+β)) & β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆ(0[,]+β)) & β’ (π β πΆ β (πΊ tsums πΉ)) β β’ (π β πΆ = βͺ (πΊ tsums πΉ)) | ||
Theorem | cntzun 31684 | The centralizer of a union is the intersection of the centralizers. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
β’ π΅ = (Baseβπ) & β’ π = (Cntzβπ) β β’ ((π β π΅ β§ π β π΅) β (πβ(π βͺ π)) = ((πβπ) β© (πβπ))) | ||
Theorem | cntzsnid 31685 | The centralizer of the identity element is the whole base set. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
β’ π΅ = (Baseβπ) & β’ π = (Cntzβπ) & β’ 0 = (0gβπ) β β’ (π β Mnd β (πβ{ 0 }) = π΅) | ||
Theorem | cntrcrng 31686 | The center of a ring is a commutative ring. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
β’ π = (π βΎs (Cntrβ(mulGrpβπ ))) β β’ (π β Ring β π β CRing) | ||
Syntax | comnd 31687 | Extend class notation with the class of all right ordered monoids. |
class oMnd | ||
Syntax | cogrp 31688 | Extend class notation with the class of all right ordered groups. |
class oGrp | ||
Definition | df-omnd 31689* | Define class of all right ordered monoids. An ordered monoid is a monoid with a total ordering compatible with its operation. It is possible to use this definition also for left ordered monoids, by applying it to (oppgβπ). (Contributed by Thierry Arnoux, 13-Mar-2018.) |
β’ oMnd = {π β Mnd β£ [(Baseβπ) / π£][(+gβπ) / π][(leβπ) / π](π β Toset β§ βπ β π£ βπ β π£ βπ β π£ (πππ β (πππ)π(πππ)))} | ||
Definition | df-ogrp 31690 | Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
β’ oGrp = (Grp β© oMnd) | ||
Theorem | isomnd 31691* | A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & β’ β€ = (leβπ) β β’ (π β oMnd β (π β Mnd β§ π β Toset β§ βπ β π΅ βπ β π΅ βπ β π΅ (π β€ π β (π + π) β€ (π + π)))) | ||
Theorem | isogrp 31692 | A (left-)ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
β’ (πΊ β oGrp β (πΊ β Grp β§ πΊ β oMnd)) | ||
Theorem | ogrpgrp 31693 | A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.) |
β’ (πΊ β oGrp β πΊ β Grp) | ||
Theorem | omndmnd 31694 | A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
β’ (π β oMnd β π β Mnd) | ||
Theorem | omndtos 31695 | A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
β’ (π β oMnd β π β Toset) | ||
Theorem | omndadd 31696 | In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ + = (+gβπ) β β’ ((π β oMnd β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π + π) β€ (π + π)) | ||
Theorem | omndaddr 31697 | In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ + = (+gβπ) β β’ (((oppgβπ) β oMnd β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ π β€ π) β (π + π) β€ (π + π)) | ||
Theorem | omndadd2d 31698 | In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ + = (+gβπ) & β’ (π β π β oMnd) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β€ π) & β’ (π β π β€ π) & β’ (π β π β CMnd) β β’ (π β (π + π) β€ (π + π)) | ||
Theorem | omndadd2rd 31699 | In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.) |
β’ π΅ = (Baseβπ) & β’ β€ = (leβπ) & β’ + = (+gβπ) & β’ (π β π β oMnd) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β€ π) & β’ (π β π β€ π) & β’ (π β (oppgβπ) β oMnd) β β’ (π β (π + π) β€ (π + π)) | ||
Theorem | submomnd 31700 | A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β (π βΎs π΄) β oMnd) |
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