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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cshf1o 33101 | Condition for the cyclic shift to be a bijection. (Contributed by Thierry Arnoux, 4-Oct-2023.) |
| ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):dom 𝑊–1-1-onto→ran 𝑊) | ||
| Theorem | ressplusf 33102 | The group operation function +𝑓 of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ ⨣ = (+g‘𝐺) & ⊢ ⨣ Fn (𝐵 × 𝐵) & ⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴)) | ||
| Theorem | ressnm 33103 | The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (norm‘𝐻)) | ||
| Theorem | abvpropd2 33104 | Weaker version of abvpropd 20864. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
| ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) & ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐿)) ⇒ ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿)) | ||
| Theorem | ressprs 33105 | The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐾) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset ) | ||
| Theorem | posrasymb 33106 | A poset ordering is asymmetric. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌)) | ||
| Theorem | odutos 33107 | Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐷 = (ODual‘𝐾) ⇒ ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset) | ||
| Theorem | tlt2 33108 | In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) ⇒ ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 < 𝑋)) | ||
| Theorem | tlt3 33109 | In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) ⇒ ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋)) | ||
| Theorem | trleile 33110 | In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)) | ||
| Theorem | toslublem 33111* | Lemma for toslub 33112 and xrsclat 33150. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ≤ = (le‘𝐾) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑)))) | ||
| Theorem | toslub 33112 | In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), 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.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < )) | ||
| Theorem | tosglblem 33113* | Lemma for tosglb 33114 and xrsclat 33150. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ≤ = (le‘𝐾) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)))) | ||
| Theorem | tosglb 33114 | Same theorem as toslub 33112, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < )) | ||
| Theorem | clatp0cl 33115 | The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0.‘𝑊) ⇒ ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵) | ||
| Theorem | clatp1cl 33116 | The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 1 = (1.‘𝑊) ⇒ ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵) | ||
| Syntax | cmnt 33117 | Extend class notation with monotone functions. |
| class Monot | ||
| Syntax | cmgc 33118 | Extend class notation with the monotone Galois connection. |
| class MGalConn | ||
| Definition | df-mnt 33119* | Define a monotone function between two ordered sets. (Contributed by Thierry Arnoux, 20-Apr-2024.) |
| ⊢ Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))}) | ||
| Definition | df-mgc 33120* | Define monotone Galois connections. See mgcval 33126 for an expanded version. (Contributed by Thierry Arnoux, 20-Apr-2024.) |
| ⊢ MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))}) | ||
| Theorem | mntoval 33121* | Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))}) | ||
| Theorem | ismnt 33122* | Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))) | ||
| Theorem | ismntd 33123 | Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ 𝐶) & ⊢ (𝜑 → 𝑊 ∈ 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌)) | ||
| Theorem | mntf 33124 | A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴⟶𝐵) | ||
| Theorem | mgcoval 33125* | Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))}) | ||
| Theorem | mgcval 33126* |
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 𝑊, 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 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) | ||
| Theorem | mgcf1 33127 | The lower adjoint 𝐹 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | ||
| Theorem | mgcf2 33128 | The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) ⇒ ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | ||
| Theorem | mgccole1 33129 | An inequality for the kernel operator 𝐺 ∘ 𝐹. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋))) | ||
| Theorem | mgccole2 33130 | Inequality for the closure operator (𝐹 ∘ 𝐺) of the Galois connection 𝐻. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌) | ||
| Theorem | mgcmnt1 33131 | The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌)) | ||
| Theorem | mgcmnt2 33132 | The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≲ 𝑌) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌)) | ||
| Theorem | mgcmntco 33133* | A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ 𝐶 = (Base‘𝑋) & ⊢ < = (le‘𝑋) & ⊢ (𝜑 → 𝑋 ∈ Proset ) & ⊢ (𝜑 → 𝐾 ∈ (𝑉Monot𝑋)) & ⊢ (𝜑 → 𝐿 ∈ (𝑊Monot𝑋)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦))) | ||
| Theorem | dfmgc2lem 33134* | Lemma for dfmgc2, backwards direction. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) & ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥))) & ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢) ⇒ ⊢ (𝜑 → 𝐹𝐻𝐺) | ||
| Theorem | dfmgc2 33135* | Alternate definition of the monotone Galois connection. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))))) | ||
| Theorem | mgcmnt1d 33136 | Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.) |
| ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) | ||
| Theorem | mgcmnt2d 33137 | Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.) |
| ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ (𝜑 → 𝑉 ∈ Proset ) & ⊢ (𝜑 → 𝑊 ∈ Proset ) & ⊢ (𝜑 → 𝐹𝐻𝐺) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉)) | ||
| Theorem | mgccnv 33138 | The inverse Galois connection is the Galois connection of the dual orders. (Contributed by Thierry Arnoux, 26-Apr-2024.) |
| ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ 𝑀 = ((ODual‘𝑊)MGalConn(ODual‘𝑉)) ⇒ ⊢ ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝐹𝐻𝐺 ↔ 𝐺𝑀𝐹)) | ||
| Theorem | pwrssmgc 33139* | 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 𝑌 ↦ (◡𝐹 “ 𝑛)) & ⊢ 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚}) & ⊢ 𝑉 = (toInc‘𝒫 𝑌) & ⊢ 𝑊 = (toInc‘𝒫 𝑋) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) ⇒ ⊢ (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻) | ||
| Theorem | mgcf1olem1 33140 | Property of a Galois connection, lemma for mgcf1o 33142. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
| ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ Poset) & ⊢ (𝜑 → 𝑊 ∈ Poset) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)) | ||
| Theorem | mgcf1olem2 33141 | Property of a Galois connection, lemma for mgcf1o 33142. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
| ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ Poset) & ⊢ (𝜑 → 𝑊 ∈ Poset) & ⊢ (𝜑 → 𝐹𝐻𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌)) | ||
| Theorem | mgcf1o 33142 | Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.) |
| ⊢ 𝐻 = (𝑉MGalConn𝑊) & ⊢ 𝐴 = (Base‘𝑉) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ≤ = (le‘𝑉) & ⊢ ≲ = (le‘𝑊) & ⊢ (𝜑 → 𝑉 ∈ Poset) & ⊢ (𝜑 → 𝑊 ∈ Poset) & ⊢ (𝜑 → 𝐹𝐻𝐺) ⇒ ⊢ (𝜑 → (𝐹 ↾ ran 𝐺) Isom ≤ , ≲ (ran 𝐺, ran 𝐹)) | ||
| Axiom | ax-xrssca 33143 | 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 = (Scalar‘ℝ*𝑠) | ||
| Axiom | ax-xrsvsca 33144 | 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.) |
| ⊢ ·e = ( ·𝑠 ‘ℝ*𝑠) | ||
| Theorem | xrs0 33145 | The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13249 and df-xrs 17515), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
| ⊢ 0 = (0g‘ℝ*𝑠) | ||
| Theorem | xrslt 33146 | The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ < = (lt‘ℝ*𝑠) | ||
| Theorem | xrsinvgval 33147 | The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13249 and df-xrs 17515), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
| ⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵) | ||
| Theorem | xrsmulgzz 33148 | The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
| ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) | ||
| Theorem | xrstos 33149 | The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.) |
| ⊢ ℝ*𝑠 ∈ Toset | ||
| Theorem | xrsclat 33150 | The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.) |
| ⊢ ℝ*𝑠 ∈ CLat | ||
| Theorem | xrsp0 33151 | 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.‘ℝ*𝑠) | ||
| Theorem | xrsp1 33152 | The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) |
| ⊢ +∞ = (1.‘ℝ*𝑠) | ||
| Theorem | xrge00 33153 | The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
| ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | ||
| Theorem | xrge0mulgnn0 33154 | The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
| ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵)) | ||
| Theorem | xrge0addass 33155 | Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.) |
| ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) | ||
| Theorem | xrge0addgt0 33156 | The sum of nonnegative and positive numbers is positive. See addgtge0 11672. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
| ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵)) | ||
| Theorem | xrge0adddir 33157 | Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.) |
| ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶))) | ||
| Theorem | xrge0adddi 33158 | Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ·e (𝐴 +𝑒 𝐵)) = ((𝐶 ·e 𝐴) +𝑒 (𝐶 ·e 𝐵))) | ||
| Theorem | xrge0npcan 33159 | Extended nonnegative real version of npcan 11436. (Contributed by Thierry Arnoux, 9-Jun-2017.) |
| ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) | ||
| Theorem | fsumrp0cl 33160* | Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) | ||
| Theorem | mndcld 33161 | Closure of the operation of a monoid. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) | ||
| Theorem | mndassd 33162 | A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
| Theorem | mndlrinv 33163 | In a monoid, if an element 𝑋 has both a left inverse 𝑀 and a right inverse 𝑁, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → (𝑀 + 𝑋) = 0 ) & ⊢ (𝜑 → (𝑋 + 𝑁) = 0 ) ⇒ ⊢ (𝜑 → 𝑀 = 𝑁) | ||
| Theorem | mndlrinvb 33164* | In a monoid, if an element has both a left-inverse and a right-inverse, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))) | ||
| Theorem | mndlactf1 33165* | If an element 𝑋 of a monoid 𝐸 is right-invertible, with inverse 𝑌, then its left-translation 𝐹 is injective. See also grplactf1o 19069. Remark in chapter I. of [BourbakiAlg1] p. 17 . (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎)) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑌 + 𝑋) = 0 ) ⇒ ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵) | ||
| Theorem | mndlactfo 33166* | An element 𝑋 of a monoid 𝐸 is left-invertible iff its left-translation 𝐹 is surjective. See also grplactf1o 19069. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎)) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 )) | ||
| Theorem | mndractf1 33167* | If an element 𝑋 of a monoid 𝐸 is right-invertible, with inverse 𝑌, then its left-translation 𝐺 is injective. See also grplactf1o 19069. Remark in chapter I. of [BourbakiAlg1] p. 17 . (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 + 𝑌) = 0 ) ⇒ ⊢ (𝜑 → 𝐺:𝐵–1-1→𝐵) | ||
| Theorem | mndractfo 33168* | An element 𝑋 of a monoid 𝐸 is right-invertible iff its right-translation 𝐺 is surjective. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) | ||
| Theorem | mndlactf1o 33169* | An element 𝑋 of a monoid 𝐸 is invertible iff its left-translation 𝐹 is bijective. See also grplactf1o 19069. Remark in chapter I. of [BourbakiAlg1] p. 17. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ 𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎)) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))) | ||
| Theorem | mndractf1o 33170* | An element 𝑋 of a monoid 𝐸 is invertible iff its right-translation 𝐺 is bijective. See also mndlactf1o 33169. Remark in chapter I. of [BourbakiAlg1] p. 17 . (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐸) & ⊢ 0 = (0g‘𝐸) & ⊢ + = (+g‘𝐸) & ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋)) & ⊢ (𝜑 → 𝐸 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 ))) | ||
| Theorem | cmn4d 33171 | Commutative/associative law for commutative monoids. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) | ||
| Theorem | cmn246135 33172 | Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33411. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 𝑉 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉)))) | ||
| Theorem | cmn145236 33173 | Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33411. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 𝑉 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑋 + (𝑈 + 𝑉)) + (𝑌 + (𝑍 + 𝑊)))) | ||
| Theorem | submcld 33174 | Submonoids are closed under the monoid operation. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆) | ||
| Theorem | abliso 33175 | 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)) → 𝑁 ∈ Abel) | ||
| Theorem | lmhmghmd 33176 | A module homomorphism is a group homomorphism. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | ||
| Theorem | mhmimasplusg 33177 | Value of the operation of the surjective image. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑊 = (𝐹 “s 𝑉) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ 𝐶 = (Base‘𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐵–onto→𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑉 MndHom 𝑊)) & ⊢ + = (+g‘𝑉) & ⊢ ⨣ = (+g‘𝑊) ⇒ ⊢ (𝜑 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)) = (𝐹‘(𝑋 + 𝑌))) | ||
| Theorem | lmhmimasvsca 33178 | Value of the scalar product of the surjective image of a module. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝑊 = (𝐹 “s 𝑉) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ 𝐶 = (Base‘𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹:𝐵–onto→𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑉 LMHom 𝑊)) & ⊢ · = ( ·𝑠 ‘𝑉) & ⊢ × = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘(Scalar‘𝑉)) ⇒ ⊢ (𝜑 → (𝑋 × (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) | ||
| Theorem | grpidcld 33179 | The identity element of a group belongs to the group. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 0 ∈ 𝐵) | ||
| Theorem | grpinvinvd 33180 | Double inverse law for groups. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋) | ||
| Theorem | grpsubcld 33181 | Closure of group subtraction. (Contributed by Thierry Arnoux, 3-Aug-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝐵) | ||
| Theorem | subgcld 33182 | A subgroup is closed under group operation. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆) | ||
| Theorem | subgsubcld 33183 | A subgroup is closed under group subtraction. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑆) | ||
| Theorem | subgmulgcld 33184 | Closure of the group multiple within a subgroup. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝑅)) & ⊢ (𝜑 → 𝑍 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑍 · 𝐴) ∈ 𝑆) | ||
| Theorem | ressmulgnn0d 33185 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
| ⊢ (𝜑 → (𝐺 ↾s 𝐴) = 𝐻) & ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻)) & ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋)) | ||
| Theorem | ablcomd 33186 | An abelian group operation is commutative, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
| Theorem | gsumsubg 33187 | The group sum in a subgroup is the same as the group sum. (Contributed by Thierry Arnoux, 28-May-2023.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
| Theorem | gsumsra 33188 | 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 ‘𝑅)‘𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝑈) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 Σg 𝐹) = (𝐴 Σg 𝐹)) | ||
| Theorem | gsummpt2co 33189* | Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐷) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))))) | ||
| Theorem | gsummpt2d 33190* | Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19995. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
| ⊢ Ⅎ𝑧𝐶 & ⊢ Ⅎ𝑦𝜑 & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷) & ⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))) | ||
| Theorem | lmodvslmhm 33191* | Scalar multiplication in a left module by a fixed element is a group homomorphism. (Contributed by Thierry Arnoux, 12-Jun-2023.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐾 ↦ (𝑥 · 𝑌)) ∈ (𝐹 GrpHom 𝑊)) | ||
| Theorem | gsumvsmul1 33192* | Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc1 20343, since every ring is a left module over itself. (Contributed by Thierry Arnoux, 12-Jun-2023.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑆 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ LMod) & ⊢ (𝜑 → 𝑆 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑆 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) | ||
| Theorem | gsummptres 33193* | Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐷)) → 𝐶 = 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) | ||
| Theorem | gsummptres2 33194* | Extend a finite group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) | ||
| Theorem | gsummptfsres 33195* | Extend a finitely supported group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) | ||
| Theorem | gsummptf1od 33196* | Re-index a finite group sum using a bijection. (Contributed by Thierry Arnoux, 29-Mar-2018.) |
| ⊢ Ⅎ𝑥𝐻 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑥 = 𝐸) → 𝐶 = 𝐻) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) | ||
| Theorem | gsummptrev 33197* | Revert ordering in a group sum. See also gsumwrev 19389. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑋 ∈ 𝐵) & ⊢ (((𝜑 ∧ 𝑙 ∈ (0...𝑁)) ∧ 𝑘 = (𝑁 − 𝑙)) → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑋)) = (𝑀 Σg (𝑙 ∈ (0...𝑁) ↦ 𝑌))) | ||
| Theorem | gsummptp1 33198* | Reindex a zero-based sum as a one-base sum. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝑁)) → 𝑌 ∈ 𝐵) & ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑙 = (𝑘 + 1)) → 𝑌 = 𝑋) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ (0..^𝑁) ↦ 𝑋)) = (𝑅 Σg (𝑙 ∈ (1...𝑁) ↦ 𝑌))) | ||
| Theorem | gsummptfzsplitra 33199* | Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the right. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑁) → 𝑌 = 𝑋) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (𝑀..^𝑁) ↦ 𝑌)) + 𝑋)) | ||
| Theorem | gsummptfzsplitla 33200* | Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the left. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑌 = 𝑋) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝑌)) = (𝑋 + (𝐺 Σg (𝑘 ∈ ((𝑀 + 1)...𝑁) ↦ 𝑌)))) | ||
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