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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mgmhmrcl 44401 | Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) | ||
Theorem | submgmrcl 44402 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) | ||
Theorem | ismgmhm 44403* | 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 44404 | 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 44405* | 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 44406 | 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 44407 | 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 44408 | The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀)) | ||
Theorem | issubmgm 44409* | 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 44410 | 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 44411* | 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 44412 | Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 ⊆ 𝐵) | ||
Theorem | submgmid 44413 | Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmcl 44414 | Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.) |
⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | submgmmgm 44415 | 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 44416 | The base set of a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻)) | ||
Theorem | subsubmgm 44417 | 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 44418 | 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 44419 | One direction of resmgmhm2b 44420. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) | ||
Theorem | resmgmhm2b 44420 | 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 44421 | 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 44422 | 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 44423 | 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 44424 | Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵)) | ||
Theorem | ismhm0 44425 | 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 44426 | Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.) |
⊢ (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆)) | ||
Theorem | opmpoismgm 44427* | A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑀 ∈ Mgm) | ||
Theorem | copissgrp 44428* | A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑀 ∈ Smgrp) | ||
Theorem | copisnmnd 44429* | A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 1 < (♯‘𝐵)) ⇒ ⊢ (𝜑 → 𝑀 ∉ Mnd) | ||
Theorem | 0nodd 44430* | 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 0 ∉ 𝑂 | ||
Theorem | 1odd 44431* | 1 is an odd integer. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 1 ∈ 𝑂 | ||
Theorem | 2nodd 44432* | 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} ⇒ ⊢ 2 ∉ 𝑂 | ||
Theorem | oddibas 44433* | Lemma 1 for oddinmgm 44435: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ 𝑂 = (Base‘𝑀) | ||
Theorem | oddiadd 44434* | Lemma 2 for oddinmgm 44435: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ + = (+g‘𝑀) | ||
Theorem | oddinmgm 44435* | The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 44568, and even a non-unital ring, see 2zrng 44559. (Contributed by AV, 3-Feb-2020.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} & ⊢ 𝑀 = (ℂfld ↾s 𝑂) ⇒ ⊢ 𝑀 ∉ Mgm | ||
Theorem | nnsgrpmgm 44436 | The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∈ Mgm | ||
Theorem | nnsgrp 44437 | The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∈ Smgrp | ||
Theorem | nnsgrpnmnd 44438 | The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝑀 = (ℂfld ↾s ℕ) ⇒ ⊢ 𝑀 ∉ Mnd | ||
Theorem | nn0mnd 44439 | The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂfld ↾s ℕ0). (Contributed by AV, 27-Dec-2023.) |
⊢ 𝑀 = {〈(Base‘ndx), ℕ0〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ 𝑀 ∈ Mnd | ||
Theorem | gsumsplit2f 44440* | Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) |
⊢ Ⅎ𝑘𝜑 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) | ||
Theorem | gsumdifsndf 44441* | Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.) |
⊢ Ⅎ𝑘𝑌 & ⊢ Ⅎ𝑘𝜑 & ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) | ||
Theorem | gsumfsupp 44442 | A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝐼 = (𝐹 supp 0 ) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹)) | ||
With df-mpo 7140, binary operations are defined by a rule, and with df-ov 7138, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆⟶𝑆. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, we call binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 internal binary operations, see df-intop 44459. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 44460. Therefore, a "binary operation on a set 𝑆 " according to Wikipedia is a "closed internal binary operation" in our terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 44460 ). Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 44460 and df-assintop 44461), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obey these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves. In the following, an alternate definition df-cllaw 44446 for an internal binary operation is provided, which does not require function-ness, but only closure. Therefore, this definition could be used as binary operation (Slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 44454, or for an alternate definition df-mgm2 44479 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups). | ||
In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi. | ||
Syntax | ccllaw 44443 | Extend class notation for the closure law. |
class clLaw | ||
Syntax | casslaw 44444 | Extend class notation for the associative law. |
class assLaw | ||
Syntax | ccomlaw 44445 | Extend class notation for the commutative law. |
class comLaw | ||
Definition | df-cllaw 44446* | The closure law for binary operations, see definitions of laws A0. and M0. in section 1.1 of [Hall] p. 1, or definition 1 in [BourbakiAlg1] p. 1: the value of a binary operation applied to two operands of a given sets is an element of this set. By this definition, the closure law is expressed as binary relation: a binary operation is related to a set by clLaw if the closure law holds for this binary operation regarding this set. Note that the binary operation needs not to be a function. (Contributed by AV, 7-Jan-2020.) |
⊢ clLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚} | ||
Definition | df-comlaw 44447* | The commutative law for binary operations, see definitions of laws A2. and M2. in section 1.1 of [Hall] p. 1, or definition 8 in [BourbakiAlg1] p. 7: the value of a binary operation applied to two operands equals the value of a binary operation applied to the two operands in reversed order. By this definition, the commutative law is expressed as binary relation: a binary operation is related to a set by comLaw if the commutative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by AV, 7-Jan-2020.) |
⊢ comLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)} | ||
Definition | df-asslaw 44448* | The associative law for binary operations, see definitions of laws A1. and M1. in section 1.1 of [Hall] p. 1, or definition 5 in [BourbakiAlg1] p. 4: the value of a binary operation applied the value of the binary operation applied to two operands and a third operand equals the value of the binary operation applied to the first operand and the value of the binary operation applied to the second and third operand. By this definition, the associative law is expressed as binary relation: a binary operation is related to a set by assLaw if the associative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.) |
⊢ assLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))} | ||
Theorem | iscllaw 44449* | The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.) |
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀)) | ||
Theorem | iscomlaw 44450* | The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.) |
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) | ||
Theorem | clcllaw 44451 | Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.) |
⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀) | ||
Theorem | isasslaw 44452* | The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.) |
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) | ||
Theorem | asslawass 44453* | Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.) |
⊢ ( ⚬ assLaw 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))) | ||
Theorem | mgmplusgiopALT 44454 | Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀)) | ||
Theorem | sgrpplusgaopALT 44455 | Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺)) | ||
In this subsection, "internal binary operations" obeying different laws are defined. | ||
Syntax | cintop 44456 | Extend class notation with class of internal (binary) operations for a set. |
class intOp | ||
Syntax | cclintop 44457 | Extend class notation with class of closed operations for a set. |
class clIntOp | ||
Syntax | cassintop 44458 | Extend class notation with class of associative operations for a set. |
class assIntOp | ||
Definition | df-intop 44459* | Function mapping a set to the class of all internal (binary) operations for this set. (Contributed by AV, 20-Jan-2020.) |
⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚))) | ||
Definition | df-clintop 44460 | Function mapping a set to the class of all closed (internal binary) operations for this set, see definition in section 1.2 of [Hall] p. 2, definition in section I.1 of [Bruck] p. 1, or definition 1 in [BourbakiAlg1] p. 1, where it is called "a law of composition". (Contributed by AV, 20-Jan-2020.) |
⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)) | ||
Definition | df-assintop 44461* | Function mapping a set to the class of all associative (closed internal binary) operations for this set, see definition 5 in [BourbakiAlg1] p. 4, where it is called "an associative law of composition". (Contributed by AV, 20-Jan-2020.) |
⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚}) | ||
Theorem | intopval 44462 | The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀))) | ||
Theorem | intop 44463 | An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁) | ||
Theorem | clintopval 44464 | The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) | ||
Theorem | assintopval 44465* | The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) | ||
Theorem | assintopmap 44466* | The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.) |
⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}) | ||
Theorem | isclintop 44467 | The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) | ||
Theorem | clintop 44468 | A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀) | ||
Theorem | assintop 44469 | An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀)) | ||
Theorem | isassintop 44470* | The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))) | ||
Theorem | clintopcllaw 44471 | The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀) | ||
Theorem | assintopcllaw 44472 | The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀) | ||
Theorem | assintopasslaw 44473 | The associative low holds for a associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ assLaw 𝑀) | ||
Theorem | assintopass 44474* | An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))) | ||
Syntax | cmgm2 44475 | Extend class notation with class of all magmas. |
class MgmALT | ||
Syntax | ccmgm2 44476 | Extend class notation with class of all commutative magmas. |
class CMgmALT | ||
Syntax | csgrp2 44477 | Extend class notation with class of all semigroups. |
class SGrpALT | ||
Syntax | ccsgrp2 44478 | Extend class notation with class of all commutative semigroups. |
class CSGrpALT | ||
Definition | df-mgm2 44479 | A magma is a set equipped with a closed operation. Definition 1 of [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by AV, 6-Jan-2020.) |
⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)} | ||
Definition | df-cmgm2 44480 | A commutative magma is a magma with a commutative operation. Definition 8 of [BourbakiAlg1] p. 7. (Contributed by AV, 20-Jan-2020.) |
⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)} | ||
Definition | df-sgrp2 44481 | A semigroup is a magma with an associative operation. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4, or of a semigroup in section 1.3 of [Hall] p. 7. (Contributed by AV, 6-Jan-2020.) |
⊢ SGrpALT = {𝑔 ∈ MgmALT ∣ (+g‘𝑔) assLaw (Base‘𝑔)} | ||
Definition | df-csgrp2 44482 | A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.) |
⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g‘𝑔) comLaw (Base‘𝑔)} | ||
Theorem | ismgmALT 44483 | The predicate "is a magma". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵)) | ||
Theorem | iscmgmALT 44484 | The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵)) | ||
Theorem | issgrpALT 44485 | The predicate "is a semigroup". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵)) | ||
Theorem | iscsgrpALT 44486 | The predicate "is a commutative semigroup". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ⚬ = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵)) | ||
Theorem | mgm2mgm 44487 | Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.) |
⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm) | ||
Theorem | sgrp2sgrp 44488 | Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.) |
⊢ (𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp) | ||
Theorem | idfusubc0 44489* | The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.) |
⊢ 𝑆 = (𝐶 ↾cat 𝐽) & ⊢ 𝐼 = (idfunc‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))〉) | ||
Theorem | idfusubc 44490* | The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.) |
⊢ 𝑆 = (𝐶 ↾cat 𝐽) & ⊢ 𝐼 = (idfunc‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = 〈( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))〉) | ||
Theorem | inclfusubc 44491* | The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.) |
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) & ⊢ 𝑆 = (𝐶 ↾cat 𝐽) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺) | ||
Theorem | lmod0rng 44492 | If the scalar ring of a module is the zero ring, the module is the zero module, i.e. the base set of the module is the singleton consisting of the identity element only. (Contributed by AV, 17-Apr-2019.) |
⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g‘𝑀)}) | ||
Theorem | nzrneg1ne0 44493 | The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.) |
⊢ (𝑅 ∈ NzRing → ((invg‘𝑅)‘(1r‘𝑅)) ≠ (0g‘𝑅)) | ||
Theorem | 0ringdif 44494 | A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 })) | ||
Theorem | 0ringbas 44495 | The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 𝐵 = { 0 }) | ||
Theorem | 0ring1eq0 44496 | In a zero ring, a ring which is not a nonzero ring, the unit equals the zero element. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 1 = 0 ) | ||
Theorem | nrhmzr 44497 | There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.) |
⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅) | ||
According to Wikipedia, "... in abstract algebra, a rng (or pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 6-Jan-2020). | ||
Syntax | crng 44498 | Extend class notation with class of all non-unital rings. |
class Rng | ||
Definition | df-rng0 44499* | Define class of all (non-unital) rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.) |
⊢ Rng = {𝑓 ∈ Abel ∣ ((mulGrp‘𝑓) ∈ Smgrp ∧ [(Base‘𝑓) / 𝑏][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))} | ||
Theorem | isrng 44500* | The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ Smgrp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))) |
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