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Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabvpropd2 31601 Weaker version of abvpropd 20224. (Contributed by Thierry Arnoux, 8-Nov-2017.)
(πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))    &   (πœ‘ β†’ (+gβ€˜πΎ) = (+gβ€˜πΏ))    &   (πœ‘ β†’ (.rβ€˜πΎ) = (.rβ€˜πΏ))    β‡’   (πœ‘ β†’ (AbsValβ€˜πΎ) = (AbsValβ€˜πΏ))
 
21.3.8.2  The opposite group
 
Theoremoppgle 31602 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppgβ€˜π‘…)    &    ≀ = (leβ€˜π‘…)    β‡’    ≀ = (leβ€˜π‘‚)
 
TheoremoppgleOLD 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.)
𝑂 = (oppgβ€˜π‘…)    &    ≀ = (leβ€˜π‘…)    β‡’    ≀ = (leβ€˜π‘‚)
 
Theoremoppglt 31604 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppgβ€˜π‘…)    &    < = (ltβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ < = (ltβ€˜π‘‚))
 
21.3.8.3  Posets
 
Theoremressprs 31605 The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐡 = (Baseβ€˜πΎ)    β‡’   ((𝐾 ∈ Proset ∧ 𝐴 βŠ† 𝐡) β†’ (𝐾 β†Ύs 𝐴) ∈ Proset )
 
Theoremoduprs 31606 Being a proset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODualβ€˜πΎ)    β‡’   (𝐾 ∈ Proset β†’ 𝐷 ∈ Proset )
 
Theoremposrasymb 31607 A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))    β‡’   ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) ↔ 𝑋 = π‘Œ))
 
Theoremresspos 31608 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 β†Ύs 𝐴) ∈ Poset)
 
Theoremresstos 31609 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Toset ∧ 𝐴 ∈ 𝑉) β†’ (𝐹 β†Ύs 𝐴) ∈ Toset)
 
Theoremodutos 31610 Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODualβ€˜πΎ)    β‡’   (𝐾 ∈ Toset β†’ 𝐷 ∈ Toset)
 
Theoremtlt2 31611 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    β‡’   ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ∨ π‘Œ < 𝑋))
 
Theoremtlt3 31612 In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    β‡’   ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 = π‘Œ ∨ 𝑋 < π‘Œ ∨ π‘Œ < 𝑋))
 
Theoremtrleile 31613 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = ((leβ€˜πΎ) ∩ (𝐡 Γ— 𝐡))    β‡’   ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ∨ π‘Œ ≀ 𝑋))
 
Theoremtoslublem 31614* Lemma for toslub 31615 and xrsclat 31653. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Toset)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &    ≀ = (leβ€˜πΎ)    β‡’   ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ((βˆ€π‘ ∈ 𝐴 𝑏 ≀ π‘Ž ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏 ≀ 𝑐 β†’ π‘Ž ≀ 𝑐)) ↔ (βˆ€π‘ ∈ 𝐴 Β¬ π‘Ž < 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (𝑏 < π‘Ž β†’ βˆƒπ‘‘ ∈ 𝐴 𝑏 < 𝑑))))
 
Theoremtoslub 31615 In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐡, < ), 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.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Toset)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    β‡’   (πœ‘ β†’ ((lubβ€˜πΎ)β€˜π΄) = sup(𝐴, 𝐡, < ))
 
Theoremtosglblem 31616* Lemma for tosglb 31617 and xrsclat 31653. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Toset)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &    ≀ = (leβ€˜πΎ)    β‡’   ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ((βˆ€π‘ ∈ 𝐴 π‘Ž ≀ 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑐 ≀ 𝑏 β†’ 𝑐 ≀ π‘Ž)) ↔ (βˆ€π‘ ∈ 𝐴 Β¬ π‘Žβ—‘ < 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (𝑏◑ < π‘Ž β†’ βˆƒπ‘‘ ∈ 𝐴 𝑏◑ < 𝑑))))
 
Theoremtosglb 31617 Same theorem as toslub 31615, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Toset)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    β‡’   (πœ‘ β†’ ((glbβ€˜πΎ)β€˜π΄) = inf(𝐴, 𝐡, < ))
 
21.3.8.4  Complete lattices
 
Theoremclatp0cl 31618 The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0.β€˜π‘Š)    β‡’   (π‘Š ∈ CLat β†’ 0 ∈ 𝐡)
 
Theoremclatp1cl 31619 The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    1 = (1.β€˜π‘Š)    β‡’   (π‘Š ∈ CLat β†’ 1 ∈ 𝐡)
 
21.3.8.5  Order Theory
 
Syntaxcmnt 31620 Extend class notation with monotone functions.
class Monot
 
Syntaxcmgc 31621 Extend class notation with the monotone Galois connection.
class MGalConn
 
Definitiondf-mnt 31622* Define a monotone function between two ordered sets. (Contributed by Thierry Arnoux, 20-Apr-2024.)
Monot = (𝑣 ∈ V, 𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œ{𝑓 ∈ ((Baseβ€˜π‘€) ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦))})
 
Definitiondf-mgc 31623* Define monotone Galois connections. See mgcval 31629 for an expanded version. (Contributed by Thierry Arnoux, 20-Apr-2024.)
MGalConn = (𝑣 ∈ V, 𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œβ¦‹(Baseβ€˜π‘€) / π‘β¦Œ{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝑏 ↑m π‘Ž) ∧ 𝑔 ∈ (π‘Ž ↑m 𝑏)) ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 ((π‘“β€˜π‘₯)(leβ€˜π‘€)𝑦 ↔ π‘₯(leβ€˜π‘£)(π‘”β€˜π‘¦)))})
 
Theoremmntoval 31624* Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    β‡’   ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝑉Monotπ‘Š) = {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))})
 
Theoremismnt 31625* Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    β‡’   ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))))
 
Theoremismntd 31626 Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   (πœ‘ β†’ 𝑉 ∈ 𝐢)    &   (πœ‘ β†’ π‘Š ∈ 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝑉Monotπ‘Š))    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐴)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    β‡’   (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))
 
Theoremmntf 31627 A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    β‡’   ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ ∧ 𝐹 ∈ (𝑉Monotπ‘Š)) β†’ 𝐹:𝐴⟢𝐡)
 
Theoremmgcoval 31628* Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    β‡’   ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝑉MGalConnπ‘Š) = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
 
Theoremmgcval 31629* Monotone Galois connection between two functions 𝐹 and 𝐺. If this relation is satisfied, 𝐹 is called the lower adjoint of 𝐺, and 𝐺 is called the upper adjoint of 𝐹.

Technically, this is implemented as an operation taking a pair of structures 𝑉 and π‘Š, expected to be posets, which gives a relation between pairs of functions 𝐹 and 𝐺.

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.)

𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    β‡’   (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))))
 
Theoremmgcf1 31630 The lower adjoint 𝐹 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    β‡’   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
 
Theoremmgcf2 31631 The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    β‡’   (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
 
Theoremmgccole1 31632 An inequality for the kernel operator 𝐺 ∘ 𝐹. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    β‡’   (πœ‘ β†’ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘‹)))
 
Theoremmgccole2 31633 Inequality for the closure operator (𝐹 ∘ 𝐺) of the Galois connection 𝐻. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (πΉβ€˜(πΊβ€˜π‘Œ)) ≲ π‘Œ)
 
Theoremmgcmnt1 31634 The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐴)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    β‡’   (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))
 
Theoremmgcmnt2 31635 The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≲ π‘Œ)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘‹) ≀ (πΊβ€˜π‘Œ))
 
Theoremmgcmntco 31636* A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   πΆ = (Baseβ€˜π‘‹)    &    < = (leβ€˜π‘‹)    &   (πœ‘ β†’ 𝑋 ∈ Proset )    &   (πœ‘ β†’ 𝐾 ∈ (𝑉Monot𝑋))    &   (πœ‘ β†’ 𝐿 ∈ (π‘ŠMonot𝑋))    β‡’   (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 (πΎβ€˜π‘₯) < (πΏβ€˜(πΉβ€˜π‘₯)) ↔ βˆ€π‘¦ ∈ 𝐡 (πΎβ€˜(πΊβ€˜π‘¦)) < (πΏβ€˜π‘¦)))
 
Theoremdfmgc2lem 31637* Lemma for dfmgc2, backwards direction. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐺:𝐡⟢𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝐡 βˆ€π‘£ ∈ 𝐡 (𝑒 ≲ 𝑣 β†’ (πΊβ€˜π‘’) ≀ (πΊβ€˜π‘£)))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ≀ (πΊβ€˜(πΉβ€˜π‘₯)))    &   ((πœ‘ ∧ 𝑒 ∈ 𝐡) β†’ (πΉβ€˜(πΊβ€˜π‘’)) ≲ 𝑒)    β‡’   (πœ‘ β†’ 𝐹𝐻𝐺)
 
Theoremdfmgc2 31638* Alternate definition of the monotone Galois connection. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   π» = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    β‡’   (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)) ∧ βˆ€π‘’ ∈ 𝐡 βˆ€π‘£ ∈ 𝐡 (𝑒 ≲ 𝑣 β†’ (πΊβ€˜π‘’) ≀ (πΊβ€˜π‘£))) ∧ (βˆ€π‘’ ∈ 𝐡 (πΉβ€˜(πΊβ€˜π‘’)) ≲ 𝑒 ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ ≀ (πΊβ€˜(πΉβ€˜π‘₯)))))))
 
Theoremmgcmnt1d 31639 Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
𝐻 = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑉Monotπ‘Š))
 
Theoremmgcmnt2d 31640 Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
𝐻 = (𝑉MGalConnπ‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Proset )    &   (πœ‘ β†’ π‘Š ∈ Proset )    &   (πœ‘ β†’ 𝐹𝐻𝐺)    β‡’   (πœ‘ β†’ 𝐺 ∈ (π‘ŠMonot𝑉))
 
Theoremmgccnv 31641 The inverse Galois connection is the Galois connection of the dual orders. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐻 = (𝑉MGalConnπ‘Š)    &   π‘€ = ((ODualβ€˜π‘Š)MGalConn(ODualβ€˜π‘‰))    β‡’   ((𝑉 ∈ Proset ∧ π‘Š ∈ Proset ) β†’ (𝐹𝐻𝐺 ↔ 𝐺𝑀𝐹))
 
Theorempwrssmgc 31642* Given a function 𝐹, exhibit a Galois connection between subsets of its domain and subsets of its range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐺 = (𝑛 ∈ 𝒫 π‘Œ ↦ (◑𝐹 β€œ 𝑛))    &   π» = (π‘š ∈ 𝒫 𝑋 ↦ {𝑦 ∈ π‘Œ ∣ (◑𝐹 β€œ {𝑦}) βŠ† π‘š})    &   π‘‰ = (toIncβ€˜π’« π‘Œ)    &   π‘Š = (toIncβ€˜π’« 𝑋)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)    β‡’   (πœ‘ β†’ 𝐺(𝑉MGalConnπ‘Š)𝐻)
 
Theoremmgcf1olem1 31643 Property of a Galois connection, lemma for mgcf1o 31645. (Contributed by Thierry Arnoux, 26-Jul-2024.)
𝐻 = (𝑉MGalConnπ‘Š)    &   π΄ = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Poset)    &   (πœ‘ β†’ π‘Š ∈ Poset)    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    β‡’   (πœ‘ β†’ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) = (πΉβ€˜π‘‹))
 
Theoremmgcf1olem2 31644 Property of a Galois connection, lemma for mgcf1o 31645. (Contributed by Thierry Arnoux, 26-Jul-2024.)
𝐻 = (𝑉MGalConnπ‘Š)    &   π΄ = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Poset)    &   (πœ‘ β†’ π‘Š ∈ Poset)    &   (πœ‘ β†’ 𝐹𝐻𝐺)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (πΊβ€˜(πΉβ€˜(πΊβ€˜π‘Œ))) = (πΊβ€˜π‘Œ))
 
Theoremmgcf1o 31645 Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.)
𝐻 = (𝑉MGalConnπ‘Š)    &   π΄ = (Baseβ€˜π‘‰)    &   π΅ = (Baseβ€˜π‘Š)    &    ≀ = (leβ€˜π‘‰)    &    ≲ = (leβ€˜π‘Š)    &   (πœ‘ β†’ 𝑉 ∈ Poset)    &   (πœ‘ β†’ π‘Š ∈ Poset)    &   (πœ‘ β†’ 𝐹𝐻𝐺)    β‡’   (πœ‘ β†’ (𝐹 β†Ύ ran 𝐺) Isom ≀ , ≲ (ran 𝐺, ran 𝐹))
 
21.3.8.6  Extended reals Structure - misc additions
 
Axiomax-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.)
ℝfld = (Scalarβ€˜β„*𝑠)
 
Axiomax-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·e = ( ·𝑠 β€˜β„*𝑠)
 
Theoremxrs0 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.)
0 = (0gβ€˜β„*𝑠)
 
Theoremxrslt 31649 The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)
< = (ltβ€˜β„*𝑠)
 
Theoremxrsinvgval 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.)
(𝐡 ∈ ℝ* β†’ ((invgβ€˜β„*𝑠)β€˜π΅) = -𝑒𝐡)
 
Theoremxrsmulgzz 31651 The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(.gβ€˜β„*𝑠)𝐡) = (𝐴 Β·e 𝐡))
 
Theoremxrstos 31652 The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
ℝ*𝑠 ∈ Toset
 
Theoremxrsclat 31653 The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
ℝ*𝑠 ∈ CLat
 
Theoremxrsp0 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.)
-∞ = (0.β€˜β„*𝑠)
 
Theoremxrsp1 31655 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
+∞ = (1.β€˜β„*𝑠)
 
Theoremressmulgnn 31656 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.)
𝐻 = (𝐺 β†Ύs 𝐴)    &   π΄ βŠ† (Baseβ€˜πΊ)    &    βˆ— = (.gβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    β‡’   ((𝑁 ∈ β„• ∧ 𝑋 ∈ 𝐴) β†’ (𝑁(.gβ€˜π»)𝑋) = (𝑁 βˆ— 𝑋))
 
Theoremressmulgnn0 31657 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
𝐻 = (𝐺 β†Ύs 𝐴)    &   π΄ βŠ† (Baseβ€˜πΊ)    &    βˆ— = (.gβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    &   (0gβ€˜πΊ) = (0gβ€˜π»)    β‡’   ((𝑁 ∈ β„•0 ∧ 𝑋 ∈ 𝐴) β†’ (𝑁(.gβ€˜π»)𝑋) = (𝑁 βˆ— 𝑋))
 
21.3.8.7  The extended nonnegative real numbers commutative monoid
 
Theoremxrge0base 31658 The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(0[,]+∞) = (Baseβ€˜(ℝ*𝑠 β†Ύs (0[,]+∞)))
 
Theoremxrge00 31659 The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
0 = (0gβ€˜(ℝ*𝑠 β†Ύs (0[,]+∞)))
 
Theoremxrge0plusg 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.)
+𝑒 = (+gβ€˜(ℝ*𝑠 β†Ύs (0[,]+∞)))
 
Theoremxrge0le 31661 The "less than or equal to" relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.)
≀ = (leβ€˜(ℝ*𝑠 β†Ύs (0[,]+∞)))
 
Theoremxrge0mulgnn0 31662 The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ β„•0 ∧ 𝐡 ∈ (0[,]+∞)) β†’ (𝐴(.gβ€˜(ℝ*𝑠 β†Ύs (0[,]+∞)))𝐡) = (𝐴 Β·e 𝐡))
 
Theoremxrge0addass 31663 Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐡 ∈ (0[,]+∞) ∧ 𝐢 ∈ (0[,]+∞)) β†’ ((𝐴 +𝑒 𝐡) +𝑒 𝐢) = (𝐴 +𝑒 (𝐡 +𝑒 𝐢)))
 
Theoremxrge0addgt0 31664 The sum of nonnegative and positive numbers is positive. See addgtge0 11577. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(((𝐴 ∈ (0[,]+∞) ∧ 𝐡 ∈ (0[,]+∞)) ∧ 0 < 𝐴) β†’ 0 < (𝐴 +𝑒 𝐡))
 
Theoremxrge0adddir 31665 Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐡 ∈ (0[,]+∞) ∧ 𝐢 ∈ (0[,]+∞)) β†’ ((𝐴 +𝑒 𝐡) Β·e 𝐢) = ((𝐴 Β·e 𝐢) +𝑒 (𝐡 Β·e 𝐢)))
 
Theoremxrge0adddi 31666 Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐡 ∈ (0[,]+∞) ∧ 𝐢 ∈ (0[,]+∞)) β†’ (𝐢 Β·e (𝐴 +𝑒 𝐡)) = ((𝐢 Β·e 𝐴) +𝑒 (𝐢 Β·e 𝐡)))
 
Theoremxrge0npcan 31667 Extended nonnegative real version of npcan 11344. (Contributed by Thierry Arnoux, 9-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐡 ∈ (0[,]+∞) ∧ 𝐡 ≀ 𝐴) β†’ ((𝐴 +𝑒 -𝑒𝐡) +𝑒 𝐡) = 𝐴)
 
Theoremfsumrp0cl 31668* Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ (0[,)+∞))    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 ∈ (0[,)+∞))
 
21.3.9  Algebra
 
21.3.9.1  Monoids Homomorphisms
 
Theoremabliso 31669 The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)
((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) β†’ 𝑁 ∈ Abel)
 
21.3.9.2  Finitely supported group sums - misc additions
 
Theoremgsumsubg 31670 The group sum in a subgroup is the same as the group sum. (Contributed by Thierry Arnoux, 28-May-2023.)
𝐻 = (𝐺 β†Ύs 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐡 ∈ (SubGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (𝐻 Ξ£g 𝐹))
 
Theoremgsumsra 31671 The group sum in a subring algebra is the same as the ring's group sum. (Contributed by Thierry Arnoux, 28-May-2023.)
𝐴 = ((subringAlg β€˜π‘…)β€˜π΅)    &   (πœ‘ β†’ 𝐹 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 ∈ π‘Š)    &   (πœ‘ β†’ 𝐡 βŠ† (Baseβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝑅 Ξ£g 𝐹) = (𝐴 Ξ£g 𝐹))
 
Theoremgsummpt2co 31672* Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐸 ∈ 𝑉)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐢 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐷 ∈ 𝐸)    &   πΉ = (π‘₯ ∈ 𝐴 ↦ 𝐷)    β‡’   (πœ‘ β†’ (π‘Š Ξ£g (π‘₯ ∈ 𝐴 ↦ 𝐢)) = (π‘Š Ξ£g (𝑦 ∈ 𝐸 ↦ (π‘Š Ξ£g (π‘₯ ∈ (◑𝐹 β€œ {𝑦}) ↦ 𝐢)))))
 
Theoremgsummpt2d 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.)
Ⅎ𝑧𝐢    &   β„²π‘¦πœ‘    &   π΅ = (Baseβ€˜π‘Š)    &   (π‘₯ = βŸ¨π‘¦, π‘§βŸ© β†’ 𝐢 = 𝐷)    &   (πœ‘ β†’ Rel 𝐴)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ π‘Š ∈ CMnd)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐢 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘Š Ξ£g (π‘₯ ∈ 𝐴 ↦ 𝐢)) = (π‘Š Ξ£g (𝑦 ∈ dom 𝐴 ↦ (π‘Š Ξ£g (𝑧 ∈ (𝐴 β€œ {𝑦}) ↦ 𝐷)))))
 
Theoremlmodvslmhm 31674* Scalar multiplication in a left module by a fixed element is a group homomorphism. (Contributed by Thierry Arnoux, 12-Jun-2023.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘₯ ∈ 𝐾 ↦ (π‘₯ Β· π‘Œ)) ∈ (𝐹 GrpHom π‘Š))
 
Theoremgsumvsmul1 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.)
𝐡 = (Baseβ€˜π‘…)    &   π‘† = (Scalarβ€˜π‘…)    &   πΎ = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ LMod)    &   (πœ‘ β†’ 𝑆 ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ 𝐾)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ (𝑋 Β· π‘Œ))) = ((𝑆 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) Β· π‘Œ))
 
Theoremgsummptres 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.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐢 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴 βˆ– 𝐷)) β†’ 𝐢 = 0 )    β‡’   (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐴 ↦ 𝐢)) = (𝐺 Ξ£g (π‘₯ ∈ (𝐴 ∩ 𝐷) ↦ 𝐢)))
 
Theoremgsummptres2 31677* Extend a finite group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 22-Jun-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   ((πœ‘ ∧ π‘₯ ∈ (𝐴 βˆ– 𝑆)) β†’ π‘Œ = 0 )    &   (πœ‘ β†’ 𝑆 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐴)    β‡’   (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ 𝐴 ↦ π‘Œ)) = (𝐺 Ξ£g (π‘₯ ∈ 𝑆 ↦ π‘Œ)))
 
Theoremgsumzresunsn 31678 Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   π‘Œ = (πΉβ€˜π‘‹)    &   (πœ‘ β†’ 𝐹:𝐢⟢𝐡)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ 𝑋 ∈ 𝐢)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ (𝐹 β€œ (𝐴 βˆͺ {𝑋})) βŠ† (π‘β€˜(𝐹 β€œ (𝐴 βˆͺ {𝑋}))))    β‡’   (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ (𝐴 βˆͺ {𝑋}))) = ((𝐺 Ξ£g (𝐹 β†Ύ 𝐴)) + π‘Œ))
 
Theoremgsumpart 31679* Express a group sum as a double sum, grouping along a (possibly infinite) partition. (Contributed by Thierry Arnoux, 22-Jun-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ CMnd)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ π‘Š)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐹 finSupp 0 )    &   (πœ‘ β†’ Disj π‘₯ ∈ 𝑋 𝐢)    &   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝑋 𝐢 = 𝐴)    β‡’   (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘₯ ∈ 𝑋 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝐢)))))
 
Theoremgsumhashmul 31680* Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    Β· = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ CMnd)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐹 finSupp 0 )    β‡’   (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((β™―β€˜(◑𝐹 β€œ {π‘₯})) Β· π‘₯))))
 
Theoremxrge0tsmsd 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.)
𝐺 = (ℝ*𝑠 β†Ύs (0[,]+∞))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐴⟢(0[,]+∞))    &   (πœ‘ β†’ 𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑠))), ℝ*, < ))    β‡’   (πœ‘ β†’ (𝐺 tsums 𝐹) = {𝑆})
 
Theoremxrge0tsmsbi 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.)
𝐺 = (ℝ*𝑠 β†Ύs (0[,]+∞))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐴⟢(0[,]+∞))    β‡’   (πœ‘ β†’ (𝐢 ∈ (𝐺 tsums 𝐹) ↔ 𝐢 = βˆͺ (𝐺 tsums 𝐹)))
 
Theoremxrge0tsmseq 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.)
𝐺 = (ℝ*𝑠 β†Ύs (0[,]+∞))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐴⟢(0[,]+∞))    &   (πœ‘ β†’ 𝐢 ∈ (𝐺 tsums 𝐹))    β‡’   (πœ‘ β†’ 𝐢 = βˆͺ (𝐺 tsums 𝐹))
 
21.3.9.3  Centralizers and centers - misc additions
 
Theoremcntzun 31684 The centralizer of a union is the intersection of the centralizers. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐡 = (Baseβ€˜π‘€)    &   π‘ = (Cntzβ€˜π‘€)    β‡’   ((𝑋 βŠ† 𝐡 ∧ π‘Œ βŠ† 𝐡) β†’ (π‘β€˜(𝑋 βˆͺ π‘Œ)) = ((π‘β€˜π‘‹) ∩ (π‘β€˜π‘Œ)))
 
Theoremcntzsnid 31685 The centralizer of the identity element is the whole base set. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐡 = (Baseβ€˜π‘€)    &   π‘ = (Cntzβ€˜π‘€)    &    0 = (0gβ€˜π‘€)    β‡’   (𝑀 ∈ Mnd β†’ (π‘β€˜{ 0 }) = 𝐡)
 
Theoremcntrcrng 31686 The center of a ring is a commutative ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝑍 = (𝑅 β†Ύs (Cntrβ€˜(mulGrpβ€˜π‘…)))    β‡’   (𝑅 ∈ Ring β†’ 𝑍 ∈ CRing)
 
21.3.9.4  Totally ordered monoids and groups
 
Syntaxcomnd 31687 Extend class notation with the class of all right ordered monoids.
class oMnd
 
Syntaxcogrp 31688 Extend class notation with the class of all right ordered groups.
class oGrp
 
Definitiondf-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 (oppgβ€˜π‘€). (Contributed by Thierry Arnoux, 13-Mar-2018.)
oMnd = {𝑔 ∈ Mnd ∣ [(Baseβ€˜π‘”) / 𝑣][(+gβ€˜π‘”) / 𝑝][(leβ€˜π‘”) / 𝑙](𝑔 ∈ Toset ∧ βˆ€π‘Ž ∈ 𝑣 βˆ€π‘ ∈ 𝑣 βˆ€π‘ ∈ 𝑣 (π‘Žπ‘™π‘ β†’ (π‘Žπ‘π‘)𝑙(𝑏𝑝𝑐)))}
 
Definitiondf-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.)
oGrp = (Grp ∩ oMnd)
 
Theoremisomnd 31691* A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    β‡’   (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘Ž ≀ 𝑏 β†’ (π‘Ž + 𝑐) ≀ (𝑏 + 𝑐))))
 
Theoremisogrp 31692 A (left-)ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
 
Theoremogrpgrp 31693 A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
(𝐺 ∈ oGrp β†’ 𝐺 ∈ Grp)
 
Theoremomndmnd 31694 A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd β†’ 𝑀 ∈ Mnd)
 
Theoremomndtos 31695 A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd β†’ 𝑀 ∈ Toset)
 
Theoremomndadd 31696 In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    + = (+gβ€˜π‘€)    β‡’   ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍))
 
Theoremomndaddr 31697 In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    + = (+gβ€˜π‘€)    β‡’   (((oppgβ€˜π‘€) ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑍 + 𝑋) ≀ (𝑍 + π‘Œ))
 
Theoremomndadd2d 31698 In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ oMnd)    &   (πœ‘ β†’ π‘Š ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ 𝑍)    &   (πœ‘ β†’ π‘Œ ≀ π‘Š)    &   (πœ‘ β†’ 𝑀 ∈ CMnd)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ≀ (𝑍 + π‘Š))
 
Theoremomndadd2rd 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.)
𝐡 = (Baseβ€˜π‘€)    &    ≀ = (leβ€˜π‘€)    &    + = (+gβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ oMnd)    &   (πœ‘ β†’ π‘Š ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ 𝑍)    &   (πœ‘ β†’ π‘Œ ≀ π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘€) ∈ oMnd)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ≀ (𝑍 + π‘Š))
 
Theoremsubmomnd 31700 A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
((𝑀 ∈ oMnd ∧ (𝑀 β†Ύs 𝐴) ∈ Mnd) β†’ (𝑀 β†Ύs 𝐴) ∈ oMnd)
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