P. 188 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mhmf1o 18701	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 18702	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 18703	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 18704	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 18705	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 18706	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 18707	Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
Theorem	issubm 18708*	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 18709	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 18710	The submonoid predicate. Analogous to issubg 19036. (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 18711*	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 18712	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 19056. (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 18713	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 19057. (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 18714	Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 ⊆ 𝐵)
Theorem	submid 18715	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 18716	Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 0 = (0g‘𝑀)    ⇒   ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 0 ∈ 𝑆)
Theorem	submcl 18717	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 18718	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 18719	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 18720	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 18721	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 18722	The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
Theorem	insubm 18723	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 18724	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 18725	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 18726	One direction of resmhm2b 18727. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Theorem	resmhm2b 18727	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 18728	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 18729*	Lemma for mhmima 18730 and similar theorems, formerly part of proof for mhmima 18730. (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 18730	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 18731	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 18732	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 18733*	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 18734*	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 18735*	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 18736*	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 18737*	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 18738*	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))
10.1.8  Iterated sums in a monoid One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 17343. If order is not significant, it is simpler to use families instead.
Theorem	gsumvallem2 18739*	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 18740	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 18741*	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 18742	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 18743	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 18744	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 18745	Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 18746. (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 18746	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 18747	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 18748	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 18749	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 18750	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 18751*	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)))
10.1.9  Free monoids
Syntax	cfrmd 18752	Extend class definition with the free monoid construction.
class freeMnd
Syntax	cvrmd 18753	Extend class notation with free monoid injection.
class varFMnd
Definition	df-frmd 18754	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 18755*	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 18756	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 18757	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 18758	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 18759	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 18760	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 18761*	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 18762	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 18763	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 18764	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 18765	The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    ⇒   ⊢ ∅ = (0g‘𝑀)
Theorem	frmdsssubm 18766	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 18767	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 18768	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 18769*	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 18770*	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 18771*	Lemma for frmdup3 18772. (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 18772*	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)
10.1.9.1  Monoid of endofunctions 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 18773	Extend class notation to include the class of monoids of endofunctions.
class EndoFMnd
Definition	df-efmnd 18774*	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 19280 and symgvalstruct 19307. (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 18775*	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 18776	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 18777*	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 18778	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 18779	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 18780	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 18781	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 18782	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 18783	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 18784	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 18785*	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 18786	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 18787	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 18788	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 18789*	Lemma for symggrp 19310 and efmndsgrp 18791. Conditions for an operation to be associative. Formerly part of proof for symggrp 19310. (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 18790	The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Mgm
Theorem	efmndsgrp 18791	The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
⊢ 𝐺 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 ∈ Smgrp
Theorem	ielefmnd 18792	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 18793	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 18794	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 18795	Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.)
⊢ (EndoFMnd‘∅) ∈ Mnd
Theorem	efmndbas0 18796	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 18797	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 18798	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 18799	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 18800*	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 19319. (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)))