P. 189 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	submnd0 18801	The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element. See, for example, smndex1mnd 18945 and smndex1n0mnd 18947). (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆)) → 0 = (0g‘𝐻))
Theorem	mndinvmod 18802*	Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
Theorem	prdsplusgcl 18803	Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)
Theorem	prdsidlem 18804*	Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ + = (+g‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ 0 = (0g ∘ 𝑅)    ⇒   ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
Theorem	prdsmndd 18805	The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    ⇒   ⊢ (𝜑 → 𝑌 ∈ Mnd)
Theorem	prds0g 18806	Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    ⇒   ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌))
Theorem	pwsmnd 18807	The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Mnd)
Theorem	pws0g 18808	Zero in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 0 }) = (0g‘𝑌))
Theorem	imasmnd2 18809*	The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))    &   ⊢ (𝜑 → 0 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
Theorem	imasmnd 18810*	The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ⊢ (𝜑 → 𝑅 ∈ Mnd)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
Theorem	imasmndf1 18811	The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ 𝑈 = (𝐹 “s 𝑅)    &   ⊢ 𝑉 = (Base‘𝑅)    ⇒   ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Mnd) → 𝑈 ∈ Mnd)
Theorem	xpsmnd 18812	The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    ⇒   ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → 𝑇 ∈ Mnd)
Theorem	xpsmnd0 18813	The identity element of a binary product of monoids. (Contributed by AV, 25-Feb-2025.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    ⇒   ⊢ ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)
Theorem	mnd1 18814	The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)
Theorem	mnd1id 18815	The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼)
10.1.7  Monoid homomorphisms and submonoids
Syntax	cmhm 18816	Hom-set generator class for monoids.
class MndHom
Syntax	csubmnd 18817	Class function taking a monoid to its lattice of submonoids.
class SubMnd
Definition	df-mhm 18818*	A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))})
Definition	df-submnd 18819*	A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g‘𝑠) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)})
Theorem	ismhm 18820*	Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 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:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
Theorem	ismhmd 18821*	Deduction version of ismhm 18820. (Contributed by SN, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ + = (+g‘𝑆)    &   ⊢ ⨣ = (+g‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝑍 = (0g‘𝑇)    &   ⊢ (𝜑 → 𝑆 ∈ Mnd)    &   ⊢ (𝜑 → 𝑇 ∈ Mnd)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   ⊢ (𝜑 → (𝐹‘ 0 ) = 𝑍)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑆 MndHom 𝑇))
Theorem	mhmrcl1 18822	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
Theorem	mhmrcl2 18823	Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
Theorem	mhmf 18824	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 18825	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 18826	Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
Theorem	mhmpropd 18827*	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 18828	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 18829	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 18830	The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))
Theorem	mhmf1o 18831	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 18832	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 18833	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 18834	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 18835	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 18836	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 18837	Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
Theorem	issubm 18838*	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 18839	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 18840	The submonoid predicate. Analogous to issubg 19166. (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 18841*	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 18842	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 19186. (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 18843	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 19187. (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 18844	Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 ⊆ 𝐵)
Theorem	submid 18845	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 18846	Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 0 = (0g‘𝑀)    ⇒   ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 0 ∈ 𝑆)
Theorem	submcl 18847	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 18848	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 18849	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 18850	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 18851	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 18852	The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
Theorem	insubm 18853	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 18854	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 18855	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 18856	One direction of resmhm2b 18857. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Theorem	resmhm2b 18857	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 18858	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 18859*	Lemma for mhmima 18860 and similar theorems, formerly part of proof for mhmima 18860. (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 18860	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 18861	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 18862	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 18863*	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 18864*	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 18865*	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 18866*	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 18867*	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 18868*	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 17502. If order is not significant, it is simpler to use families instead.
Theorem	gsumvallem2 18869*	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 18870	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 18871*	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 18872	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 18873	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 18874	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 18875	Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 18876. (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 18876	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 18877	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 18878	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 18879	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 18880	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 18881*	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 18882	Extend class definition with the free monoid construction.
class freeMnd
Syntax	cvrmd 18883	Extend class notation with free monoid injection.
class varFMnd
Definition	df-frmd 18884	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 18885*	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 18886	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 18887	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 18888	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 18889	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 18890	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 18891*	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 18892	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 18893	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 18894	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 18895	The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (freeMnd‘𝐼)    ⇒   ⊢ ∅ = (0g‘𝑀)
Theorem	frmdsssubm 18896	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 18897	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 18898	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 18899*	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 18900*	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))