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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmndvcl 20601 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + 𝑌) ∈ (𝐵𝑚 𝐼))

Theoremmndvass 20602 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼) ∧ 𝑍 ∈ (𝐵𝑚 𝐼))) → ((𝑋𝑓 + 𝑌) ∘𝑓 + 𝑍) = (𝑋𝑓 + (𝑌𝑓 + 𝑍)))

Theoremmndvlid 20603 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → ((𝐼 × { 0 }) ∘𝑓 + 𝑋) = 𝑋)

Theoremmndvrid 20604 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + (𝐼 × { 0 })) = 𝑋)

Theoremgrpvlinv 20605 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → ((𝑁𝑋) ∘𝑓 + 𝑋) = (𝐼 × { 0 }))

Theoremgrpvrinv 20606 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + (𝑁𝑋)) = (𝐼 × { 0 }))

Theoremmhmvlin 20607 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (+g𝑁)       ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝐹 ∘ (𝑋𝑓 + 𝑌)) = ((𝐹𝑋) ∘𝑓 (𝐹𝑌)))

Theoremringvcl 20608 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 · 𝑌) ∈ (𝐵𝑚 𝐼))

Theoremgsumcom3 20609* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))

Theoremgsumcom3fi 20610* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))

Theoremmamucl 20611 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵𝑚 (𝑀 × 𝑃)))

Theoremmamuass 20612 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑂 × 𝑃)))    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   𝐺 = (𝑅 maMul ⟨𝑀, 𝑂, 𝑃⟩)    &   𝐻 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐼 = (𝑅 maMul ⟨𝑁, 𝑂, 𝑃⟩)       (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍)))

Theoremmamudi 20613 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((𝑋𝑓 + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘𝑓 + (𝑌𝐹𝑍)))

Theoremmamudir 20614 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(𝑌𝑓 + 𝑍)) = ((𝑋𝐹𝑌) ∘𝑓 + (𝑋𝐹𝑍)))

Theoremmamuvs1 20615 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    · = (.r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘𝑓 · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘𝑓 · (𝑌𝐹𝑍)))

Theoremmamuvs2 20616 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
(𝜑𝑅 ∈ CRing)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘𝑓 · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘𝑓 · (𝑋𝐹𝑍)))

11.2.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 20639. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection.

Syntaxcmat 20617 Syntax for the square matrix algebra.
class Mat

Definitiondf-mat 20618* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))

Theoremmatbas0pc 20619 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Theoremmatbas0 20620 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Theoremmatval 20621 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Theoremmatrcl 20622 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑋𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))

Theoremmatbas 20623 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘𝐺) = (Base‘𝐴))

Theoremmatplusg 20624 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (+g𝐺) = (+g𝐴))

Theoremmatsca 20625 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))

Theoremmatvsca 20626 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ( ·𝑠𝐺) = ( ·𝑠𝐴))

Theoremmat0 20627 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (0g𝐺) = (0g𝐴))

Theoremmatinvg 20628 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (invg𝐺) = (invg𝐴))

Theoremmat0op 20629* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁0 ))

Theoremmatsca2 20630 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 = (Scalar‘𝐴))

Theoremmatbas2 20631 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))

Theoremmatbas2i 20632 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 ∈ (𝐾𝑚 (𝑁 × 𝑁)))

Theoremmatbas2d 20633* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅𝑉)    &   ((𝜑𝑥𝑁𝑦𝑁) → 𝐶𝐾)       (𝜑 → (𝑥𝑁, 𝑦𝑁𝐶) ∈ 𝐵)

Theoremeqmat 20634* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))

Theoremmatecl 20635 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐼𝑁𝐽𝑁𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)

Theoremmatecld 20636 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝑀𝐵)       (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)

Theoremmatplusg2 20637 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))

Theoremmatvsca2 20638 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)    &   𝐶 = (𝑁 × 𝑁)       ((𝑋𝐾𝑌𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘𝑓 × 𝑌))

Theoremmatlmod 20639 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)

Theoremmatgrp 20640 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)

Theoremmatvscl 20641 Closure of the scalar multiplication in the matrix ring. (lmodvscl 19272 analog.) (Contributed by AV, 27-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑋𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)

Theoremmatsubg 20642 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (-g𝐺) = (-g𝐴))

Theoremmatplusgcell 20643 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       (((𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽)))

Theoremmatsubgcell 20644 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (-g𝐴)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) (𝐼𝑌𝐽)))

Theoremmatinvgcell 20645 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = (invg𝑅)    &   𝑊 = (invg𝐴)       ((𝑅 ∈ Ring ∧ 𝑋𝐵 ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑊𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽)))

Theoremmatvscacell 20646 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))

Theoremmatgsum 20647* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑖𝑁, 𝑗𝑁𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈)) finSupp 0 )       (𝜑 → (𝐴 Σg (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑅 Σg (𝑦𝐽𝑈))))

11.2.3  The matrix algebra

The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 20653) and an associative algebra (see matassa 20654). Additionally, theorems for the identity matrix and transposed matrices are provided.

Theoremmatmulr 20648 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → · = (.r𝐴))

Theoremmamumat1cl 20649* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       (𝜑𝐼 ∈ (𝐵𝑚 (𝑀 × 𝑀)))

Theoremmat1comp 20650* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

Theoremmamulid 20651* The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑀, 𝑁⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))       (𝜑 → (𝐼𝐹𝑋) = 𝑋)

Theoremmamurid 20652* The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑁, 𝑀, 𝑀⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑁 × 𝑀)))       (𝜑 → (𝑋𝐹𝐼) = 𝑋)

Theoremmatring 20653 Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)

Theoremmatassa 20654 Existence of the matrix algebra, see also the statement in [Lang] p. 505: "Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)

Theoremmatmulcell 20655* Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) (Revised by AV, 3-Jul-2022.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    × = (.r𝐴)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗)(.r𝑅)(𝑗𝑌𝐽)))))

TheoremmatmulcellOLD 20656* Obsolete version of matmulcell 20655 as of 3-Jul-2022 . (Contributed by AV, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝑅)    &    × = (.r𝐴)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗)(.r𝑅)(𝑗𝑌𝐽)))))

Theoremmpt2matmul 20657* Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝐴)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑁 ∈ Fin)    &   𝑋 = (𝑖𝑁, 𝑗𝑁𝐶)    &   𝑌 = (𝑖𝑁, 𝑗𝑁𝐸)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐶𝐵)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐸𝐵)    &   ((𝜑 ∧ (𝑘 = 𝑖𝑚 = 𝑗)) → 𝐷 = 𝐶)    &   ((𝜑 ∧ (𝑚 = 𝑖𝑙 = 𝑗)) → 𝐹 = 𝐸)    &   ((𝜑𝑘𝑁𝑚𝑁) → 𝐷𝑈)    &   ((𝜑𝑚𝑁𝑙𝑁) → 𝐹𝑊)       (𝜑 → (𝑋 × 𝑌) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑅 Σg (𝑚𝑁 ↦ (𝐷 · 𝐹)))))

Theoremmat1 20658* Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))

Theoremmat1ov 20659 Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   𝑈 = (1r𝐴)       (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))

Theoremmat1bas 20660 The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r‘(𝑁 Mat 𝑅))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1𝐵)

Theoremmatsc 20661* The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿𝐾) → (𝐿 · (1r𝐴)) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 )))

Theoremofco2 20662 Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
(((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))

Theoremoftpos 20663 The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.)
((𝐹𝑉𝐺𝑊) → tpos (𝐹𝑓 𝑅𝐺) = (tpos 𝐹𝑓 𝑅tpos 𝐺))

Theoremmattposcl 20664 The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → tpos 𝑀𝐵)

Theoremmattpostpos 20665 The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → tpos tpos 𝑀 = 𝑀)

Theoremmattposvs 20666 The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)       ((𝑋𝐾𝑌𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌))

Theoremmattpos1 20667 The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 )

Theoremtposmap 20668 The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
(𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)))

Theoremmamutpos 20669 Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 maMul ⟨𝑃, 𝑁, 𝑀⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → tpos (𝑋𝐹𝑌) = (tpos 𝑌𝐺tpos 𝑋))

Theoremmattposm 20670 Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝐴)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋))

Theoremmatgsumcl 20671* Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈 Σg (𝑟𝑁 ↦ (𝑟𝑀𝑟))) ∈ (Base‘𝑅))

Theoremmadetsumid 20672* The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑃 = ( I ↾ 𝑁)) → (((𝑌𝑆)‘𝑃) · (𝑈 Σg (𝑟𝑁 ↦ ((𝑃𝑟)𝑀𝑟)))) = (𝑈 Σg (𝑟𝑁 ↦ (𝑟𝑀𝑟))))

Theoremmatepmcl 20673* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → ∀𝑛𝑁 ((𝑄𝑛)𝑀𝑛) ∈ (Base‘𝑅))

Theoremmatepm2cl 20674* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → ∀𝑛𝑁 (𝑛𝑀(𝑄𝑛)) ∈ (Base‘𝑅))

Theoremmadetsmelbas 20675* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑌 = (ℤRHom‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑄𝑃) → (((𝑌𝑆)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))

Theoremmadetsmelbas2 20676* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑌 = (ℤRHom‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑄𝑃) → (((𝑌𝑆)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛𝑀(𝑄𝑛))))) ∈ (Base‘𝑅))

11.2.4  Matrices of dimension 0 and 1

As already mentioned before, and shown in mat0dimbas0 20677, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng 20681.

For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric 20698.

Theoremmat0dimbas0 20677 The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
(𝑅𝑉 → (Base‘(∅ Mat 𝑅)) = {∅})

Theoremmat0dim0 20678 The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (0g𝐴) = ∅)

Theoremmat0dimid 20679 The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (1r𝐴) = ∅)

Theoremmat0dimscm 20680 The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑋( ·𝑠𝐴)∅) = ∅)

Theoremmat0dimcrng 20681 The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → 𝐴 ∈ CRing)

Theoremmat1dimelbas 20682* A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (𝑀 ∈ (Base‘𝐴) ↔ ∃𝑟𝐵 𝑀 = {⟨𝑂, 𝑟⟩}))

Theoremmat1dimbas 20683 A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐵) → {⟨𝑂, 𝑋⟩} ∈ (Base‘𝐴))

Theoremmat1dim0 20684 The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (0g𝐴) = {⟨𝑂, (0g𝑅)⟩})

Theoremmat1dimid 20685 The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (1r𝐴) = {⟨𝑂, (1r𝑅)⟩})

Theoremmat1dimscm 20686 The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       (((𝑅 ∈ Ring ∧ 𝐸𝑉) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( ·𝑠𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r𝑅)𝑌)⟩})

Theoremmat1dimmul 20687 The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       (((𝑅 ∈ Ring ∧ 𝐸𝑉) ∧ (𝑋𝐵𝑌𝐵)) → ({⟨𝑂, 𝑋⟩} (.r𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r𝑅)𝑌)⟩})

Theoremmat1dimcrng 20688 The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ CRing ∧ 𝐸𝑉) → 𝐴 ∈ CRing)

Theoremmat1f1o 20689* There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹:𝐾1-1-onto𝐵)

Theoremmat1rhmval 20690* The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})

Theoremmat1rhmelval 20691* The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)

Theoremmat1rhmcl 20692* The value of the ring homomorphism 𝐹 is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) ∈ 𝐵)

Theoremmat1f 20693* There is a function from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹:𝐾𝐵)

Theoremmat1ghm 20694* There is a group homomorphism from the additive group of a ring to the additive group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 GrpHom 𝐴))

Theoremmat1mhm 20695* There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})    &   𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝐴)       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑀 MndHom 𝑁))

Theoremmat1rhm 20696* There is a ring homomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 RingHom 𝐴))

Theoremmat1rngiso 20697* There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴))

Theoremmat1ric 20698 A ring is isomorphic to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 30-Dec-2019.)
𝐴 = ({𝐸} Mat 𝑅)       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝑅𝑟 𝐴)

11.2.5  The subalgebras of diagonal and scalar matrices

According to Wikipedia ("Diagonal Matrix", 8-Dec-2019, https://en.wikipedia.org/wiki/Diagonal_matrix): "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices." The diagonal matrices are mentioned in [Lang] p. 576, but without giving them a dedicated definition. Furthermore, "A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple 𝜆 𝐼 of the identity matrix 𝐼. Its effect on a vector is scalar multiplication by 𝜆 [see scmatscm 20724!]". The scalar multiples of the identity matrix are mentioned in [Lang] p. 504, but without giving them a special name.

The main results of this subsection are the definitions of the sets of diagonal and scalar matrices (df-dmat 20701 and df-scmat 20702), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20712), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20731), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20734) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20732).

Syntaxcdmat 20699 Extend class notation for the algebra of diagonal matrices.
class DMat

Syntaxcscmat 20700 Extend class notation for the algebra of scalar matrices.
class ScMat

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