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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mhmrcl2 18801 | Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd) | ||
| Theorem | mhmf 18802 | A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) | ||
| Theorem | ismhm0 18803 | Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.) |
| ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝑌 = (0g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) | ||
| Theorem | mhmismgmhm 18804 | Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.) |
| ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆)) | ||
| Theorem | mhmpropd 18805* | Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) ⇒ ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀)) | ||
| Theorem | mhmlin 18806 | A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) | ||
| Theorem | mhm0 18807 | A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 0 = (0g‘𝑆) & ⊢ 𝑌 = (0g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘ 0 ) = 𝑌) | ||
| Theorem | idmhm 18808 | The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀)) | ||
| Theorem | mhmf1o 18809 | A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MndHom 𝑅))) | ||
| Theorem | mndvcl 18810 | Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + 𝑌) ∈ (𝐵 ↑m 𝐼)) | ||
| Theorem | mndvass 18811 | Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼) ∧ 𝑍 ∈ (𝐵 ↑m 𝐼))) → ((𝑋 ∘f + 𝑌) ∘f + 𝑍) = (𝑋 ∘f + (𝑌 ∘f + 𝑍))) | ||
| Theorem | mndvlid 18812 | Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝐼 × { 0 }) ∘f + 𝑋) = 𝑋) | ||
| Theorem | mndvrid 18813 | Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝐼 × { 0 })) = 𝑋) | ||
| Theorem | mhmvlin 18814 | Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+g‘𝑁) ⇒ ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝐹 ∘ (𝑋 ∘f + 𝑌)) = ((𝐹 ∘ 𝑋) ∘f ⨣ (𝐹 ∘ 𝑌))) | ||
| Theorem | submrcl 18815 | Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) | ||
| Theorem | issubm 18816* | Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))) | ||
| Theorem | issubm2 18817 | Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ 𝐻 ∈ Mnd))) | ||
| Theorem | issubmndb 18818 | The submonoid predicate. Analogous to issubg 19144. (Contributed by AV, 1-Feb-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ∈ Mnd ∧ (𝐺 ↾s 𝑆) ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆))) | ||
| Theorem | issubmd 18819* | Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂) & ⊢ (𝑧 = 0 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃)) & ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂)) ⇒ ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀)) | ||
| Theorem | mndissubm 18820 | If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. Analogous to grpissubg 19164. (Contributed by AV, 17-Feb-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (Base‘𝐻) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺))) | ||
| Theorem | resmndismnd 18821 | If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the other monoid is contained in the base set of the monoid, then the other monoid restricted to the base set of the monoid is a monoid. Analogous to resgrpisgrp 19165. (Contributed by AV, 17-Feb-2024.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (Base‘𝐻) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝐺 ↾s 𝑆) ∈ Mnd)) | ||
| Theorem | submss 18822 | Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 ⊆ 𝐵) | ||
| Theorem | submid 18823 | Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝑀)) | ||
| Theorem | subm0cl 18824 | Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 0 = (0g‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 0 ∈ 𝑆) | ||
| Theorem | submcl 18825 | Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
| Theorem | submmnd 18826 | Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝐻 ∈ Mnd) | ||
| Theorem | submbas 18827 | The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.) |
| ⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 = (Base‘𝐻)) | ||
| Theorem | subm0 18828 | Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.) |
| ⊢ 𝐻 = (𝑀 ↾s 𝑆) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 0 = (0g‘𝐻)) | ||
| Theorem | subsubm 18829 | A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.) |
| ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆))) | ||
| Theorem | 0subm 18830 | The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.) |
| ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺)) | ||
| Theorem | insubm 18831 | The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.) |
| ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀)) | ||
| Theorem | 0mhm 18832 | The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 0 = (0g‘𝑁) & ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁)) | ||
| Theorem | resmhm 18833 | Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.) |
| ⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MndHom 𝑇)) | ||
| Theorem | resmhm2 18834 | One direction of resmhm2b 18835. (Contributed by Mario Carneiro, 18-Jun-2015.) |
| ⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇)) | ||
| Theorem | resmhm2b 18835 | Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.) |
| ⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈))) | ||
| Theorem | mhmco 18836 | The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
| ⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈)) | ||
| Theorem | mhmimalem 18837* | Lemma for mhmima 18838 and similar theorems, formerly part of proof for mhmima 18838. (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 18838 | 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 18839 | 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 18840 | 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 18841* | 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 18842* | 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 18843* | 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 18844* | 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 18845* | 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 18846* | 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 17487. If order is not significant, it is simpler to use families instead. | ||
| Theorem | gsumvallem2 18847* | 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 18848 | 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 18849* | 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 18850 | 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 18851 | 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 18852 | 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 18853 | Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 18854. (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 18854 | 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 18855 | 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 18856 | 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 18857 | 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 18858 | 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 18859* | 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 18860 | Extend class definition with the free monoid construction. |
| class freeMnd | ||
| Syntax | cvrmd 18861 | Extend class notation with free monoid injection. |
| class varFMnd | ||
| Definition | df-frmd 18862 | 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 18863* | 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 18864 | 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 18865 | 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 18866 | 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 18867 | 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 18868 | 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 18869* | 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 18870 | 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 18871 | 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 18872 | 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 18873 | The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
| ⊢ 𝑀 = (freeMnd‘𝐼) ⇒ ⊢ ∅ = (0g‘𝑀) | ||
| Theorem | frmdsssubm 18874 | 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 18875 | 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 18876 | 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 18877* | 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 18878* | 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 18879* | Lemma for frmdup3 18880. (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 18880* | 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 18881 | Extend class notation to include the class of monoids of endofunctions. |
| class EndoFMnd | ||
| Definition | df-efmnd 18882* | 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 19387 and symgvalstruct 19414. (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 18883* | 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 18884 | 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 18885* | 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 18886 | 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 18887 | 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 18888 | 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 18889 | 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 18890 | 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 18891 | 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 18892 | 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 18893* | 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 18894 | 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 18895 | 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 18896 | 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 18897* | Lemma for symggrp 19418 and efmndsgrp 18899. Conditions for an operation to be associative. Formerly part of proof for symggrp 19418. (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 18898 | The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.) |
| ⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ 𝐺 ∈ Mgm | ||
| Theorem | efmndsgrp 18899 | The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.) |
| ⊢ 𝐺 = (EndoFMnd‘𝐴) ⇒ ⊢ 𝐺 ∈ Smgrp | ||
| Theorem | ielefmnd 18900 | 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)) | ||
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