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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubmnd0 17301 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.) (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐻 = (𝐺s 𝑆)       (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆)) → 0 = (0g𝐻))

Theoremprdsplusgcl 17302 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)

Theoremprdsidlem 17303* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Mnd)    &    0 = (0g𝑅)       (𝜑 → ( 0𝐵 ∧ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))

Theoremprdsmndd 17304 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)       (𝜑𝑌 ∈ Mnd)

Theoremprds0g 17305 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)       (𝜑 → (0g𝑅) = (0g𝑌))

Theorempwsmnd 17306 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉) → 𝑌 ∈ Mnd)

Theorempws0g 17307 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉) → (𝐼 × { 0 }) = (0g𝑌))

Theoremimasmnd2 17308* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅𝑊)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧))))    &   (𝜑0𝑉)    &   ((𝜑𝑥𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹𝑥))    &   ((𝜑𝑥𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹𝑥))       (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))

Theoremimasmnd 17309* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑅 ∈ Mnd)    &    0 = (0g𝑅)       (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹0 ) = (0g𝑈)))

Theoremimasmndf1 17310 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝑈 = (𝐹s 𝑅)    &   𝑉 = (Base‘𝑅)       ((𝐹:𝑉1-1𝐵𝑅 ∈ Mnd) → 𝑈 ∈ Mnd)

Theoremxpsmnd 17311 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)

Theoremmnd1 17312 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⟩}⟩}       (𝐼𝑉𝑀 ∈ Mnd)

Theoremmnd1id 17313 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⟩}⟩}       (𝐼𝑉 → (0g𝑀) = 𝐼)

10.1.6  Monoid homomorphisms and submonoids

Syntaxcmhm 17314 Hom-set generator class for monoids.
class MndHom

Syntaxcsubmnd 17315 Class function taking a monoid to its lattice of submonoids.
class SubMnd

Definitiondf-mhm 17316* 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(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})

Definitiondf-submnd 17317* 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) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})

Theoremismhm 17318* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &    0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹0 ) = 𝑌)))

Theoremmhmrcl1 17319 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)

Theoremmhmrcl2 17320 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)

Theoremmhmf 17321 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵𝐶)

Theoremmhmpropd 17322* Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))       (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))

Theoremmhmlin 17323 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Theoremmhm0 17324 A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹0 ) = 𝑌)

Theoremidmhm 17325 The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mnd → ( I ↾ 𝐵) ∈ (𝑀 MndHom 𝑀))

Theoremmhmf1o 17326 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:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MndHom 𝑅)))

Theoremsubmrcl 17327 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
(𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)

Theoremissubm 17328* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &    + = (+g𝑀)       (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Theoremissubm2 17329 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐻 = (𝑀s 𝑆)       (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆𝐻 ∈ Mnd)))

Theoremissubmd 17330* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝜒)    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)    &   (𝑧 = 0 → (𝜓𝜒))    &   (𝑧 = 𝑥 → (𝜓𝜃))    &   (𝑧 = 𝑦 → (𝜓𝜏))    &   (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))       (𝜑 → {𝑧𝐵𝜓} ∈ (SubMnd‘𝑀))

Theoremsubmss 17331 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐵 = (Base‘𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 𝑆𝐵)

Theoremsubmid 17332 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mnd → 𝐵 ∈ (SubMnd‘𝑀))

Theoremsubm0cl 17333 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
0 = (0g𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 0𝑆)

Theoremsubmcl 17334 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
+ = (+g𝑀)       ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Theoremsubmmnd 17335 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMnd‘𝑀) → 𝐻 ∈ Mnd)

Theoremsubmbas 17336 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMnd‘𝑀) → 𝑆 = (Base‘𝐻))

Theoremsubm0 17337 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
𝐻 = (𝑀s 𝑆)    &    0 = (0g𝑀)       (𝑆 ∈ (SubMnd‘𝑀) → 0 = (0g𝐻))

Theoremsubsubm 17338 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))

Theorem0mhm 17339 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0g𝑁)    &   𝐵 = (Base‘𝑀)       ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))

Theoremresmhm 17340 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MndHom 𝑇))

Theoremresmhm2 17341 One direction of resmhm2b 17342. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremresmhm2b 17342 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (𝑇s 𝑋)       ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Theoremmhmco 17343 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))

Theoremmhmima 17344 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹𝑋) ∈ (SubMnd‘𝑁))

Theoremmhmeql 17345 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 (𝐹𝐺) ∈ (SubMnd‘𝑆))

Theoremsubmacs 17346 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))

Theoremmrcmndind 17347* (( From SO's determinants branch )). TODO: Appropriate description to be added! (Contributed by SO, 14-Jul-2018.)
(𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = (𝑦 + 𝑧) → (𝜓𝜃))    &   (𝑥 = 0 → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &    0 = (0g𝑀)    &    + = (+g𝑀)    &   𝐵 = (Base‘𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝐺𝐵)    &   (𝜑𝐵 = ((mrCls‘(SubMnd‘𝑀))‘𝐺))    &   (𝜑𝜏)    &   (((𝜑𝑦𝐵𝑧𝐺) ∧ 𝜒) → 𝜃)    &   (𝜑𝐴𝐵)       (𝜑𝜂)

Theoremprdspjmhm 17348* 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)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝑋)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐴𝐼)       (𝜑 → (𝑥𝐵 ↦ (𝑥𝐴)) ∈ (𝑌 MndHom (𝑅𝐴)))

Theorempwspjmhm 17349* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Mnd ∧ 𝐼𝑉𝐴𝐼) → (𝑥𝐵 ↦ (𝑥𝐴)) ∈ (𝑌 MndHom 𝑅))

Theorempwsdiagmhm 17350* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ Mnd ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 MndHom 𝑌))

Theorempwsco1mhm 17351* Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑌 = (𝑅s 𝐴)    &   𝑍 = (𝑅s 𝐵)    &   𝐶 = (Base‘𝑍)    &   (𝜑𝑅 ∈ Mnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))

Theorempwsco2mhm 17352* Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑌 = (𝑅s 𝐴)    &   𝑍 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹 ∈ (𝑅 MndHom 𝑆))       (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍))

10.1.7  Ordered 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 16084. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem2 17353* 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 = (0g𝐺)    &    + = (+g𝐺)    &   𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}       (𝐺 ∈ Mnd → 𝑂 = { 0 })

Theoremgsumsubm 17354 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝐹:𝐴𝑆)    &   𝐻 = (𝐺s 𝑆)       (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Theoremgsumz 17355* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )

Theoremgsumwsubmcl 17356 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)

Theoremgsumws1 17357 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝐵 = (Base‘𝐺)       (𝑆𝐵 → (𝐺 Σg ⟨“𝑆”⟩) = 𝑆)

Theoremgsumwcl 17358 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)

Theoremgsumccat 17359 Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))

Theoremgsumws2 17360 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)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑆𝐵𝑇𝐵) → (𝐺 Σg ⟨“𝑆𝑇”⟩) = (𝑆 + 𝑇))

Theoremgsumccatsn 17361 Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑍𝐵) → (𝐺 Σg (𝑊 ++ ⟨“𝑍”⟩)) = ((𝐺 Σg 𝑊) + 𝑍))

Theoremgsumspl 17362 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)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝑆 ∈ Word 𝐵)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(#‘𝑆)))    &   (𝜑𝑋 ∈ Word 𝐵)    &   (𝜑𝑌 ∈ Word 𝐵)    &   (𝜑 → (𝑀 Σg 𝑋) = (𝑀 Σg 𝑌))       (𝜑 → (𝑀 Σg (𝑆 splice ⟨𝐹, 𝑇, 𝑋⟩)) = (𝑀 Σg (𝑆 splice ⟨𝐹, 𝑇, 𝑌⟩)))

Theoremgsumwmhm 17363 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 (𝐻𝑊)))

Theoremgsumwspan 17364* 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 ∧ 𝐺𝐵) → (𝐾𝐺) = ran (𝑤 ∈ Word 𝐺 ↦ (𝑀 Σg 𝑤)))

10.1.8  Free monoids

Syntaxcfrmd 17365 Extend class definition with the free monoid construction.
class freeMnd

Syntaxcvrmd 17366 Extend class notation with free monoid injection.
class varFMnd

Definitiondf-frmd 17367 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))⟩})

Definitiondf-vrmd 17368* 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 ↦ (𝑗𝑖 ↦ ⟨“𝑗”⟩))

Theoremfrmdval 17369 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   (𝐼𝑉𝐵 = Word 𝐼)    &    + = ( ++ ↾ (𝐵 × 𝐵))       (𝐼𝑉𝑀 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩})

Theoremfrmdbas 17370 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)       (𝐼𝑉𝐵 = Word 𝐼)

Theoremfrmdelbas 17371 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)       (𝑋𝐵𝑋 ∈ Word 𝐼)

Theoremfrmdplusg 17372 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)    &    + = (+g𝑀)        + = ( ++ ↾ (𝐵 × 𝐵))

Theoremfrmdadd 17373 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋 ++ 𝑌))

Theoremvrmdfval 17374* The canonical injection from the generating set 𝐼 to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       (𝐼𝑉𝑈 = (𝑗𝐼 ↦ ⟨“𝑗”⟩))

Theoremvrmdval 17375 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = ⟨“𝐴”⟩)

Theoremvrmdf 17376 The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
𝑈 = (varFMnd𝐼)       (𝐼𝑉𝑈:𝐼⟶Word 𝐼)

Theoremfrmdmnd 17377 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑀 = (freeMnd‘𝐼)       (𝐼𝑉𝑀 ∈ Mnd)

Theoremfrmd0 17378 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (freeMnd‘𝐼)       ∅ = (0g𝑀)

Theoremfrmdsssubm 17379 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)       ((𝐼𝑉𝐽𝐼) → Word 𝐽 ∈ (SubMnd‘𝑀))

Theoremfrmdgsum 17380 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 = (varFMnd𝐼)       ((𝐼𝑉𝑊 ∈ Word 𝐼) → (𝑀 Σg (𝑈𝑊)) = 𝑊)

Theoremfrmdss2 17381 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 = (varFMnd𝐼)       ((𝐼𝑉𝐽𝐼𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈𝐽) ⊆ 𝐴 ↔ Word 𝐽𝐴))

Theoremfrmdup1 17382* 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 (𝐴𝑥)))    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑋)    &   (𝜑𝐴:𝐼𝐵)       (𝜑𝐸 ∈ (𝑀 MndHom 𝐺))

Theoremfrmdup2 17383* 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 (𝐴𝑥)))    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐼𝑋)    &   (𝜑𝐴:𝐼𝐵)    &   𝑈 = (varFMnd𝐼)    &   (𝜑𝑌𝐼)       (𝜑 → (𝐸‘(𝑈𝑌)) = (𝐴𝑌))

Theoremfrmdup3lem 17384* Lemma for frmdup3 17385. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝐺)    &   𝑈 = (varFMnd𝐼)       (((𝐺 ∈ Mnd ∧ 𝐼𝑉𝐴:𝐼𝐵) ∧ (𝐹 ∈ (𝑀 MndHom 𝐺) ∧ (𝐹𝑈) = 𝐴)) → 𝐹 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴𝑥))))

Theoremfrmdup3 17385* 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 = (varFMnd𝐼)       ((𝐺 ∈ Mnd ∧ 𝐼𝑉𝐴:𝐼𝐵) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚𝑈) = 𝐴)

10.1.9  Examples and counterexamples for magmas, semigroups and monoids

Theoremmgm2nsgrplem1 17386* Lemma 1 for mgm2nsgrp 17390: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17235). (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))       ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)

Theoremmgm2nsgrplem2 17387* Lemma 2 for mgm2nsgrp 17390. (Contributed by AV, 27-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))    &    = (+g𝑀)       ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)

Theoremmgm2nsgrplem3 17388* Lemma 3 for mgm2nsgrp 17390. (Contributed by AV, 28-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))    &    = (+g𝑀)       ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)

Theoremmgm2nsgrplem4 17389* Lemma 4 for mgm2nsgrp 17390: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))       ((#‘𝑆) = 2 → 𝑀 ∉ SGrp)

Theoremmgm2nsgrp 17390* A small magma (with two elements) which is not a semigroup. (Contributed by AV, 28-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))       ((#‘𝑆) = 2 → (𝑀 ∈ Mgm ∧ 𝑀 ∉ SGrp))

Theoremsgrp2nmndlem1 17391* Lemma 1 for sgrp2nmnd 17398: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17235). (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((𝐴𝑉𝐵𝑊) → 𝑀 ∈ Mgm)

Theoremsgrp2nmndlem2 17392* Lemma 2 for sgrp2nmnd 17398. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)

Theoremsgrp2nmndlem3 17393* Lemma 3 for sgrp2nmnd 17398. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)

Theoremsgrp2rid2 17394* A small semigroup (with two elements) with two right identities which are different if 𝐴𝐵. (Contributed by AV, 10-Feb-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((𝐴𝑉𝐵𝑊) → ∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦)

Theoremsgrp2rid2ex 17395* A small semigroup (with two elements) with two right identities which are different. (Contributed by AV, 10-Feb-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))    &    = (+g𝑀)       ((#‘𝑆) = 2 → ∃𝑥𝑆𝑧𝑆𝑦𝑆 (𝑥𝑧 ∧ (𝑦 𝑥) = 𝑦 ∧ (𝑦 𝑧) = 𝑦))

Theoremsgrp2nmndlem4 17396* Lemma 4 for sgrp2nmnd 17398: M is a semigroup. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((#‘𝑆) = 2 → 𝑀 ∈ SGrp)

Theoremsgrp2nmndlem5 17397* Lemma 5 for sgrp2nmnd 17398: M is not a monoid. (Contributed by AV, 29-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((#‘𝑆) = 2 → 𝑀 ∉ Mnd)

Theoremsgrp2nmnd 17398* A small semigroup (with two elements) which is not a monoid. (Contributed by AV, 26-Jan-2020.)
𝑆 = {𝐴, 𝐵}    &   (Base‘𝑀) = 𝑆    &   (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))       ((#‘𝑆) = 2 → (𝑀 ∈ SGrp ∧ 𝑀 ∉ Mnd))

Theoremmgmnsgrpex 17399 There is a magma which is not a semigroup. (Contributed by AV, 29-Jan-2020.)
𝑚 ∈ Mgm 𝑚 ∉ SGrp

Theoremsgrpnmndex 17400 There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.)
𝑚 ∈ SGrp 𝑚 ∉ Mnd

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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