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Type | Label | Description |
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Statement | ||
Theorem | mhmimalem 18701* | Lemma for mhmima 18702 and similar theorems, formerly part of proof for mhmima 18702. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.) |
⊢ (𝜑 → 𝐹 ∈ (𝑀 MndHom 𝑁)) & ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑀)) & ⊢ (𝜑 → ⊕ = (+g‘𝑀)) & ⊢ (𝜑 → + = (+g‘𝑁)) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧 ⊕ 𝑥) ∈ 𝑋) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋)) | ||
Theorem | mhmima 18702 | The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.) (Proof shortened by AV, 8-Mar-2025.) |
⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMnd‘𝑁)) | ||
Theorem | mhmeql 18703 | The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆)) | ||
Theorem | submacs 18704 | Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵)) | ||
Theorem | mndind 18705* | Induction in a monoid. In this theorem, 𝜓(𝑥) is the "generic" proposition to be be proved (the first four hypotheses tell its values at y, y+z, 0, A respectively). The two induction hypotheses mndind.i1 and mndind.i2 tell that it is true at 0, that if it is true at y then it is true at y+z (provided z is in 𝐺). The hypothesis mndind.k tells that 𝐺 is generating. (Contributed by SO, 14-Jul-2018.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 + 𝑧) → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ 0 = (0g‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐺 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 = ((mrCls‘(SubMnd‘𝑀))‘𝐺)) & ⊢ (𝜑 → 𝜏) & ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺) ∧ 𝜒) → 𝜃) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | prdspjmhm 18706* | A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) & ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom (𝑅‘𝐴))) | ||
Theorem | pwspjmhm 18707* | A projection from a structure power of a monoid to the monoid itself is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) ⇒ ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑌 MndHom 𝑅)) | ||
Theorem | pwsdiagmhm 18708* | Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 MndHom 𝑌)) | ||
Theorem | pwsco1mhm 18709* | Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐴) & ⊢ 𝑍 = (𝑅 ↑s 𝐵) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ (𝜑 → 𝑅 ∈ Mnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 MndHom 𝑌)) | ||
Theorem | pwsco2mhm 18710* | Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐴) & ⊢ 𝑍 = (𝑆 ↑s 𝐴) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 MndHom 𝑆)) ⇒ ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 MndHom 𝑍)) | ||
One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 17384. If order is not significant, it is simpler to use families instead. | ||
Theorem | gsumvallem2 18711* | Lemma for properties of the set of identities of 𝐺. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⇒ ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) | ||
Theorem | gsumsubm 18712 | Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumz 18713* | Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) | ||
Theorem | gsumwsubmcl 18714 | Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) |
⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑊 ∈ Word 𝑆) → (𝐺 Σg 𝑊) ∈ 𝑆) | ||
Theorem | gsumws1 18715 | A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) | ||
Theorem | gsumwcl 18716 | Closure of the composite of a word in a structure 𝐺. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝐺 Σg 𝑊) ∈ 𝐵) | ||
Theorem | gsumsgrpccat 18717 | Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 18718. (Contributed by AV, 26-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ (𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵) ∧ (𝑊 ≠ ∅ ∧ 𝑋 ≠ ∅)) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋))) | ||
Theorem | gsumccat 18718 | Homomorphic property of composites. Second formula in [Lang] p. 4. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 26-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋))) | ||
Theorem | gsumws2 18719 | Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵) → (𝐺 Σg 〈“𝑆𝑇”〉) = (𝑆 + 𝑇)) | ||
Theorem | gsumccatsn 18720 | Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐺 Σg (𝑊 ++ 〈“𝑍”〉)) = ((𝐺 Σg 𝑊) + 𝑍)) | ||
Theorem | gsumspl 18721 | The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝑆 ∈ Word 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (0...𝑇)) & ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆))) & ⊢ (𝜑 → 𝑋 ∈ Word 𝐵) & ⊢ (𝜑 → 𝑌 ∈ Word 𝐵) & ⊢ (𝜑 → (𝑀 Σg 𝑋) = (𝑀 Σg 𝑌)) ⇒ ⊢ (𝜑 → (𝑀 Σg (𝑆 splice 〈𝐹, 𝑇, 𝑋〉)) = (𝑀 Σg (𝑆 splice 〈𝐹, 𝑇, 𝑌〉))) | ||
Theorem | gsumwmhm 18722 | Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝐻 ∈ (𝑀 MndHom 𝑁) ∧ 𝑊 ∈ Word 𝐵) → (𝐻‘(𝑀 Σg 𝑊)) = (𝑁 Σg (𝐻 ∘ 𝑊))) | ||
Theorem | gsumwspan 18723* | The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (mrCls‘(SubMnd‘𝑀)) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝐺 ⊆ 𝐵) → (𝐾‘𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤))) | ||
Syntax | cfrmd 18724 | Extend class definition with the free monoid construction. |
class freeMnd | ||
Syntax | cvrmd 18725 | Extend class notation with free monoid injection. |
class varFMnd | ||
Definition | df-frmd 18726 | Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx), Word 𝑖〉, 〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉}) | ||
Definition | df-vrmd 18727* | Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ 〈“𝑗”〉)) | ||
Theorem | frmdval 18728 | Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) & ⊢ + = ( ++ ↾ (𝐵 × 𝐵)) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉}) | ||
Theorem | frmdbas 18729 | The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼) | ||
Theorem | frmdelbas 18730 | An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝐼) | ||
Theorem | frmdplusg 18731 | The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) (Proof shortened by AV, 6-Nov-2024.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ + = ( ++ ↾ (𝐵 × 𝐵)) | ||
Theorem | frmdadd 18732 | Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ++ 𝑌)) | ||
Theorem | vrmdfval 18733* | The canonical injection from the generating set 𝐼 to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ 〈“𝑗”〉)) | ||
Theorem | vrmdval 18734 | The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = 〈“𝐴”〉) | ||
Theorem | vrmdf 18735 | The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼) | ||
Theorem | frmdmnd 18736 | A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑀 = (freeMnd‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) | ||
Theorem | frmd0 18737 | The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) ⇒ ⊢ ∅ = (0g‘𝑀) | ||
Theorem | frmdsssubm 18738 | The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑀 = (freeMnd‘𝐼) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → Word 𝐽 ∈ (SubMnd‘𝑀)) | ||
Theorem | frmdgsum 18739 | Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐼) → (𝑀 Σg (𝑈 ∘ 𝑊)) = 𝑊) | ||
Theorem | frmdss2 18740 | A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of 𝐽 is Word 𝐽". (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴)) | ||
Theorem | frmdup1 18741* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺)) | ||
Theorem | frmdup2 18742* | The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐵) & ⊢ 𝑈 = (varFMnd‘𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) | ||
Theorem | frmdup3lem 18743* | Lemma for frmdup3 18744. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ (𝐹 ∈ (𝑀 MndHom 𝐺) ∧ (𝐹 ∘ 𝑈) = 𝐴)) → 𝐹 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))) | ||
Theorem | frmdup3 18744* | Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑈 = (varFMnd‘𝐼) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚 ∘ 𝑈) = 𝐴) | ||
According to Wikipedia ("Endomorphism", 25-Jan-2024, https://en.wikipedia.org/wiki/Endomorphism) "An endofunction is a function whose domain is equal to its codomain.". An endofunction is sometimes also called "self-mapping" (see https://www.wikidata.org/wiki/Q1691962) or "self-map" (see https://mathworld.wolfram.com/Self-Map.html), in German "Selbstabbildung" (see https://de.wikipedia.org/wiki/Selbstabbildung). | ||
Syntax | cefmnd 18745 | Extend class notation to include the class of monoids of endofunctions. |
class EndoFMnd | ||
Definition | df-efmnd 18746* | Define the monoid of endofunctions on set 𝑥. We represent the monoid as the set of functions from 𝑥 to itself ((𝑥 ↑m 𝑥)) under function composition, and topologize it as a function space assuming the set is discrete. Analogous to the former definition of SymGrp, see df-symg 19229 and symgvalstruct 19258. (Contributed by AV, 25-Jan-2024.) |
⊢ EndoFMnd = (𝑥 ∈ V ↦ ⦋(𝑥 ↑m 𝑥) / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))〉}) | ||
Theorem | efmnd 18747* | The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (𝐴 ↑m 𝐴) & ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) & ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉}) | ||
Theorem | efmndbas 18748 | The base set of the monoid of endofunctions on class 𝐴. (Contributed by AV, 25-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 = (𝐴 ↑m 𝐴) | ||
Theorem | efmndbasabf 18749* | The base set of the monoid of endofunctions on class 𝐴 is the set of functions from 𝐴 into itself. (Contributed by AV, 29-Mar-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴⟶𝐴} | ||
Theorem | elefmndbas 18750 | Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐴)) | ||
Theorem | elefmndbas2 18751 | Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 29-Mar-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐴)) | ||
Theorem | efmndbasf 18752 | Elements in the monoid of endofunctions on 𝐴 are functions from 𝐴 into itself. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴⟶𝐴) | ||
Theorem | efmndhash 18753 | The monoid of endofunctions on 𝑛 objects has cardinality 𝑛↑𝑛. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ Fin → (♯‘𝐵) = ((♯‘𝐴)↑(♯‘𝐴))) | ||
Theorem | efmndbasfi 18754 | The monoid of endofunctions on a finite set 𝐴 is finite. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ Fin → 𝐵 ∈ Fin) | ||
Theorem | efmndfv 18755 | The function value of an endofunction. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐴) | ||
Theorem | efmndtset 18756 | The topology of the monoid of endofunctions on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺)) | ||
Theorem | efmndplusg 18757* | The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) | ||
Theorem | efmndov 18758 | The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌)) | ||
Theorem | efmndcl 18759 | The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | ||
Theorem | efmndtopn 18760 | The topology of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝑋) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺)) | ||
Theorem | symggrplem 18761* | Lemma for symggrp 19262 and efmndsgrp 18763. Conditions for an operation to be associative. Formerly part of proof for symggrp 19262. (Contributed by AV, 28-Jan-2024.) |
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ∘ 𝑦)) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
Theorem | efmndmgm 18762 | The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ 𝐺 ∈ Mgm | ||
Theorem | efmndsgrp 18763 | The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ 𝐺 ∈ Smgrp | ||
Theorem | ielefmnd 18764 | The identity function restricted to a set 𝐴 is an element of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) | ||
Theorem | efmndid 18765 | The identity function restricted to a set 𝐴 is the identity element of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺)) | ||
Theorem | efmndmnd 18766 | The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd) | ||
Theorem | efmnd0nmnd 18767 | Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.) |
⊢ (EndoFMnd‘∅) ∈ Mnd | ||
Theorem | efmndbas0 18768 | The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 31-Mar-2024.) |
⊢ (Base‘(EndoFMnd‘∅)) = {∅} | ||
Theorem | efmnd1hash 18769 | The monoid of endofunctions on a singleton has cardinality 1. (Contributed by AV, 27-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐴 = {𝐼} ⇒ ⊢ (𝐼 ∈ 𝑉 → (♯‘𝐵) = 1) | ||
Theorem | efmnd1bas 18770 | The monoid of endofunctions on a singleton consists of the identity only. (Contributed by AV, 31-Jan-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐴 = {𝐼} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{〈𝐼, 𝐼〉}}) | ||
Theorem | efmnd2hash 18771 | The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐴 = {𝐼, 𝐽} ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 4) | ||
Theorem | submefmnd 18772* | If the base set of a monoid is contained in the base set of the monoid of endofunctions on a set 𝐴, contains the identity function and has the function composition as group operation, then its base set is a submonoid of the monoid of endofunctions on set 𝐴. Analogous to pgrpsubgsymg 19271. (Contributed by AV, 17-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘𝐴) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ 𝐹 = (Base‘𝑆) ⇒ ⊢ (𝐴 ∈ 𝑉 → (((𝑆 ∈ Mnd ∧ 𝐹 ⊆ 𝐵 ∧ 0 ∈ 𝐹) ∧ (+g‘𝑆) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubMnd‘𝑀))) | ||
Theorem | sursubmefmnd 18773* | The set of surjective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–onto→𝐴} ∈ (SubMnd‘𝑀)) | ||
Theorem | injsubmefmnd 18774* | The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀)) | ||
Theorem | idressubmefmnd 18775 | The singleton containing only the identity function restricted to a set is a submonoid of the monoid of endofunctions on this set. (Contributed by AV, 17-Feb-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → {( I ↾ 𝐴)} ∈ (SubMnd‘𝐺)) | ||
Theorem | idresefmnd 18776 | The structure with the singleton containing only the identity function restricted to a set 𝐴 as base set and the function composition as group operation, constructed by (structure) restricting the monoid of endofunctions on 𝐴 to that singleton, is a monoid whose base set is a subset of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 17-Feb-2024.) |
⊢ 𝐺 = (EndoFMnd‘𝐴) & ⊢ 𝐸 = (𝐺 ↾s {( I ↾ 𝐴)}) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐸 ∈ Mnd ∧ (Base‘𝐸) ⊆ (Base‘𝐺))) | ||
Theorem | smndex1ibas 18777 | The modulo function 𝐼 is an endofunction on ℕ0. (Contributed by AV, 12-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) ⇒ ⊢ 𝐼 ∈ (Base‘𝑀) | ||
Theorem | smndex1iidm 18778* | The modulo function 𝐼 is idempotent. (Contributed by AV, 12-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) ⇒ ⊢ (𝐼 ∘ 𝐼) = 𝐼 | ||
Theorem | smndex1gbas 18779* | The constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) ⇒ ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) ∈ (Base‘𝑀)) | ||
Theorem | smndex1gid 18780* | The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ0 results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) ⇒ ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾)) | ||
Theorem | smndex1igid 18781* | The composition of the modulo function 𝐼 and a constant function (𝐺‘𝐾) results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) ⇒ ⊢ (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝐾)) = (𝐺‘𝐾)) | ||
Theorem | smndex1basss 18782* | The modulo function 𝐼 and the constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ⇒ ⊢ 𝐵 ⊆ (Base‘𝑀) | ||
Theorem | smndex1bas 18783* | The base set of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 12-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ (Base‘𝑆) = 𝐵 | ||
Theorem | smndex1mgm 18784* | The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a magma. (Contributed by AV, 14-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ 𝑆 ∈ Mgm | ||
Theorem | smndex1sgrp 18785* | The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a semigroup. (Contributed by AV, 14-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ 𝑆 ∈ Smgrp | ||
Theorem | smndex1mndlem 18786* | Lemma for smndex1mnd 18787 and smndex1id 18788. (Contributed by AV, 16-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋)) | ||
Theorem | smndex1mnd 18787* | The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a monoid. (Contributed by AV, 16-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ 𝑆 ∈ Mnd | ||
Theorem | smndex1id 18788* | The modulo function 𝐼 is the identity of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 16-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ 𝐼 = (0g‘𝑆) | ||
Theorem | smndex1n0mnd 18789* | The identity of the monoid 𝑀 of endofunctions on set ℕ0 is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ (0g‘𝑀) ∉ 𝐵 | ||
Theorem | nsmndex1 18790* | The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set ℕ0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) & ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) & ⊢ 𝑆 = (𝑀 ↾s 𝐵) ⇒ ⊢ 𝐵 ∉ (SubMnd‘𝑀) | ||
Theorem | smndex2dbas 18791 | The doubling function 𝐷 is an endofunction on ℕ0. (Contributed by AV, 18-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) ⇒ ⊢ 𝐷 ∈ 𝐵 | ||
Theorem | smndex2dnrinv 18792 | The doubling function 𝐷 has no right inverse in the monoid of endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) ⇒ ⊢ ∀𝑓 ∈ 𝐵 (𝐷 ∘ 𝑓) ≠ 0 | ||
Theorem | smndex2hbas 18793 | The halving functions 𝐻 are endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁)) ⇒ ⊢ 𝐻 ∈ 𝐵 | ||
Theorem | smndex2dlinvh 18794* | The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.) |
⊢ 𝑀 = (EndoFMnd‘ℕ0) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁)) ⇒ ⊢ (𝐻 ∘ 𝐷) = 0 | ||
Theorem | mgm2nsgrplem1 18795* | Lemma 1 for mgm2nsgrp 18799: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18570). (Contributed by AV, 27-Jan-2020.) |
⊢ 𝑆 = {𝐴, 𝐵} & ⊢ (Base‘𝑀) = 𝑆 & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) | ||
Theorem | mgm2nsgrplem2 18796* | Lemma 2 for mgm2nsgrp 18799. (Contributed by AV, 27-Jan-2020.) |
⊢ 𝑆 = {𝐴, 𝐵} & ⊢ (Base‘𝑀) = 𝑆 & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ⚬ 𝐴) ⚬ 𝐵) = 𝐴) | ||
Theorem | mgm2nsgrplem3 18797* | Lemma 3 for mgm2nsgrp 18799. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝑆 = {𝐴, 𝐵} & ⊢ (Base‘𝑀) = 𝑆 & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⚬ (𝐴 ⚬ 𝐵)) = 𝐵) | ||
Theorem | mgm2nsgrplem4 18798* | Lemma 4 for mgm2nsgrp 18799: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-Jan-2020.) |
⊢ 𝑆 = {𝐴, 𝐵} & ⊢ (Base‘𝑀) = 𝑆 & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) ⇒ ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Smgrp) | ||
Theorem | mgm2nsgrp 18799* | A small magma (with two elements) which is not a semigroup. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝑆 = {𝐴, 𝐵} & ⊢ (Base‘𝑀) = 𝑆 & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) ⇒ ⊢ ((♯‘𝑆) = 2 → (𝑀 ∈ Mgm ∧ 𝑀 ∉ Smgrp)) | ||
Theorem | sgrp2nmndlem1 18800* | Lemma 1 for sgrp2nmnd 18807: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18570). (Contributed by AV, 29-Jan-2020.) |
⊢ 𝑆 = {𝐴, 𝐵} & ⊢ (Base‘𝑀) = 𝑆 & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
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