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Type | Label | Description |
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Statement | ||
The symbol Σg is mostly used in the context of abelian groups. Therefore, it is usually called "group sum". It can be defined, however, in arbitrary magmas (then it should be called "iterated sum"). If the magma is not required to be commutative or associative, then the order of the summands and the order in which summations are done become important. If the magma is not unital, then one cannot define a meaningful empty sum. See Remark 2. in the comment for df-gsum 17392. | ||
Theorem | gsumvalx 18601* | Expand out the substitutions in df-gsum 17392. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)} & ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ 𝑂))) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → dom 𝐹 = 𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))))) | ||
Theorem | gsumval 18602* | Expand out the substitutions in df-gsum 17392. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)} & ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ 𝑂))) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))))) | ||
Theorem | gsumpropd 18603 | The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 18684 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumpropd2lem 18604* | Lemma for gsumpropd2 18605. (Contributed by Thierry Arnoux, 28-Jun-2017.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) & ⊢ 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})) & ⊢ 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)})) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumpropd2 18605* | A stronger version of gsumpropd 18603, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 18606. (Contributed by Thierry Arnoux, 28-Jun-2017.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsummgmpropd 18606* | A stronger version of gsumpropd 18603 if at least one of the involved structures is a magma, see gsumpropd2 18605. (Contributed by AV, 31-Jan-2020.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ Mgm) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumress 18607* | The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither 𝐺 nor 𝐻 need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 0 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumval1 18608* | Value of the group sum operation when every element being summed is an identity of 𝐺. (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)} & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑂) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) | ||
Theorem | gsum0 18609 | Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 Σg ∅) = 0 | ||
Theorem | gsumval2a 18610* | Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) & ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} & ⊢ (𝜑 → ¬ ran 𝐹 ⊆ 𝑂) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | gsumval2 18611 | Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | gsumsplit1r 18612 | Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1)))) | ||
Theorem | gsumprval 18613 | Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 = (𝑀 + 1)) & ⊢ (𝜑 → 𝐹:{𝑀, 𝑁}⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁))) | ||
Theorem | gsumpr12val 18614 | Value of the group sum operation over the pair {1, 2}. (Contributed by AV, 14-Dec-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:{1, 2}⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘1) + (𝐹‘2))) | ||
Syntax | cmgmhm 18615 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 18616 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 18617* | A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.) |
⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))}) | ||
Definition | df-submgm 18618* | A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.) |
⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | ||
Theorem | mgmhmrcl 18619 | Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) | ||
Theorem | submgmrcl 18620 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) | ||
Theorem | ismgmhm 18621* | Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))) | ||
Theorem | mgmhmf 18622 | A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) | ||
Theorem | mgmhmpropd 18623* | Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐶 ≠ ∅) & ⊢ ((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 MgmHom 𝐾) = (𝐿 MgmHom 𝑀)) | ||
Theorem | mgmhmlin 18624 | A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) | ||
Theorem | mgmhmf1o 18625 | A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MgmHom 𝑅))) | ||
Theorem | idmgmhm 18626 | The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀)) | ||
Theorem | issubmgm 18627* | Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))) | ||
Theorem | issubmgm2 18628 | Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 𝐻 ∈ Mgm))) | ||
Theorem | rabsubmgmd 18629* | Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mgm) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂) & ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃)) & ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂)) ⇒ ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmss 18630 | Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 ⊆ 𝐵) | ||
Theorem | submgmid 18631 | Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmcl 18632 | Submagmas are closed under the magma operation. (Contributed by AV, 26-Feb-2020.) |
⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | submgmmgm 18633 | Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝐻 ∈ Mgm) | ||
Theorem | submgmbas 18634 | The base set of a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻)) | ||
Theorem | subsubmgm 18635 | A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))) | ||
Theorem | resmgmhm 18636 | Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇)) | ||
Theorem | resmgmhm2 18637 | One direction of resmgmhm2b 18638. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) | ||
Theorem | resmgmhm2b 18638 | Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈))) | ||
Theorem | mgmhmco 18639 | The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.) |
⊢ ((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MgmHom 𝑈)) | ||
Theorem | mgmhmima 18640 | The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.) |
⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMgm‘𝑁)) | ||
Theorem | mgmhmeql 18641 | The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.) |
⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMgm‘𝑆)) | ||
Theorem | submgmacs 18642 | Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵)) | ||
A semigroup (Smgrp, see df-sgrp 18644) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 18644). In other words, a semigroup is an associative magma. The notion of semigroup is a generalization of that of group where the existence of an identity or inverses is not required. | ||
Syntax | csgrp 18643 | Extend class notation with class of all semigroups. |
class Smgrp | ||
Definition | df-sgrp 18644* | A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 18565), whose operation is associative. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))} | ||
Theorem | issgrp 18645* | The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) | ||
Theorem | issgrpv 18646* | The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))) | ||
Theorem | issgrpn0 18647* | The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))) | ||
Theorem | isnsgrp 18648 | A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → 𝑀 ∉ Smgrp)) | ||
Theorem | sgrpmgm 18649 | A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm) | ||
Theorem | sgrpass 18650 | A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⚬ = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))) | ||
Theorem | sgrpcl 18651 | Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⚬ = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵) | ||
Theorem | sgrp0 18652 | Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.) |
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp) | ||
Theorem | sgrp0b 18653 | The structure with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.) |
⊢ {⟨(Base‘ndx), ∅⟩, ⟨(+g‘ndx), 𝑂⟩} ∈ Smgrp | ||
Theorem | sgrp1 18654 | The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.) |
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Smgrp) | ||
Theorem | issgrpd 18655* | Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐺 ∈ Smgrp) | ||
Theorem | sgrppropd 18656* | If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.) |
⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp)) | ||
Theorem | prdsplusgsgrpcl 18657 | Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) | ||
Theorem | prdssgrpd 18658 | The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Smgrp) ⇒ ⊢ (𝜑 → 𝑌 ∈ Smgrp) | ||
According to Wikipedia ("Monoid", https://en.wikipedia.org/wiki/Monoid, 6-Feb-2020,) "In abstract algebra [...] a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity.". In the following, monoids are defined in the second way (as semigroups with identity), see df-mnd 18660, whereas many authors define magmas in the first way (as algebraic structure with a single associative binary operation and an identity element, i.e. without the need of a definition for/knowledge about semigroups), see ismnd 18662. See, for example, the definition in [Lang] p. 3: "A monoid is a set G, with a law of composition which is associative, and having a unit element". | ||
Syntax | cmnd 18659 | Extend class notation with class of all monoids. |
class Mnd | ||
Definition | df-mnd 18660* | A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 18667), whose operation is associative (see mndass 18668) and has a two-sided neutral element (see mndid 18669), see also ismnd 18662. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)} | ||
Theorem | ismnddef 18661* | The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))) | ||
Theorem | ismnd 18662* | The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 18667), whose operation is associative (so, a semigroup, see also mndass 18668) and has a two-sided neutral element (see mndid 18669). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))) | ||
Theorem | isnmnd 18663* | A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd) | ||
Theorem | sgrpidmnd 18664* | A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 18854 and pwmndid 18853), but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd) | ||
Theorem | mndsgrp 18665 | A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp) | ||
Theorem | mndmgm 18666 | A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) |
⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm) | ||
Theorem | mndcl 18667 | Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Proof shortened by AV, 8-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | ||
Theorem | mndass 18668 | A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
Theorem | mndid 18669* | A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) | ||
Theorem | mndideu 18670* | The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)) | ||
Theorem | mnd32g 18671 | Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) | ||
Theorem | mnd12g 18672 | Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) ⇒ ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) | ||
Theorem | mnd4g 18673 | Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) | ||
Theorem | mndidcl 18674 | The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) | ||
Theorem | mndbn0 18675 | The base set of a monoid is not empty. Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → 𝐵 ≠ ∅) | ||
Theorem | hashfinmndnn 18676 | A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ) | ||
Theorem | mndplusf 18677 | The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⨣ = (+𝑓‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵) | ||
Theorem | mndlrid 18678 | A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)) | ||
Theorem | mndlid 18679 | The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) | ||
Theorem | mndrid 18680 | The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) | ||
Theorem | ismndd 18681* | Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 0 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) ⇒ ⊢ (𝜑 → 𝐺 ∈ Mnd) | ||
Theorem | mndpfo 18682 | The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⨣ = (+𝑓‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵) | ||
Theorem | mndfo 18683 | The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵) | ||
Theorem | mndpropd 18684* | If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)) | ||
Theorem | mndprop 18685 | If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) |
⊢ (Base‘𝐾) = (Base‘𝐿) & ⊢ (+g‘𝐾) = (+g‘𝐿) ⇒ ⊢ (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd) | ||
Theorem | issubmnd 18686* | Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆) → (𝐻 ∈ Mnd ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)) | ||
Theorem | ress0g 18687 | 0g is unaffected by restriction. This is a bit more generic than submnd0 18688. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝑆)) | ||
Theorem | submnd0 18688 | 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 18827 and smndex1n0mnd 18829). (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 18689* | 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 18690 | 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 18691* | 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 18692 | 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 18693 | 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 18694 | 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 18695 | 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 18696* | 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 18697* | 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 18698 | 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 18699 | 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 18700 | 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)⟩) |
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