P. 203 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grpvlinv 20201	Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> ( ( N o. X ) oF .+ X ) = ( I X. { .0. } ) )
Theorem	grpvrinv 20202	Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- N = ( invg ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> ( X oF .+ ( N o. X ) ) = ( I X. { .0. } ) )
Theorem	mhmvlin 20203	Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` M )   &    |- .+ = ( +g ` M )   &    |- .+^ = ( +g ` N )   =>    |- ( ( F e. ( M MndHom N ) /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> ( F o. ( X oF .+ Y ) ) = ( ( F o. X ) oF .+^ ( F o. Y ) ) )
Theorem	ringvcl 20204	Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ X e. ( B ^m I ) /\ Y e. ( B ^m I ) ) -> ( X oF .x. Y ) e. ( B ^m I ) )
Theorem	gsumcom3 20205*	A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. V )   &    |- ( ph -> C e. W )   &    |- ( ( ph /\ ( j e. A /\ k e. C ) ) -> X e. B )   &    |- ( ph -> U e. Fin )   &    |- ( ( ph /\ ( ( j e. A /\ k e. C ) /\ -. j U k ) ) -> X = .0. )   =>    |- ( ph -> ( G gsumg ( j e. A |-> ( G gsumg ( k e. C |-> X ) ) ) ) = ( G gsumg ( k e. C |-> ( G gsumg ( j e. A |-> X ) ) ) ) )
Theorem	gsumcom3fi 20206*	A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> C e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. C ) ) -> X e. B )   =>    |- ( ph -> ( G gsumg ( j e. A |-> ( G gsumg ( k e. C |-> X ) ) ) ) = ( G gsumg ( k e. C |-> ( G gsumg ( j e. A |-> X ) ) ) ) )
Theorem	mamucl 20207	Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
|- B = ( Base ` R )   &    |- ( ph -> R e. Ring )   &    |- F = ( R maMul <. M , N , P >. )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> P e. Fin )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Y e. ( B ^m ( N X. P ) ) )   =>    |- ( ph -> ( X F Y ) e. ( B ^m ( M X. P ) ) )
Theorem	mamuass 20208	Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> O e. Fin )   &    |- ( ph -> P e. Fin )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Y e. ( B ^m ( N X. O ) ) )   &    |- ( ph -> Z e. ( B ^m ( O X. 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 >. )   =>    |- ( ph -> ( ( X F Y ) G Z ) = ( X H ( Y I Z ) ) )
Theorem	mamudi 20209	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 )   &    |- ( ph -> R e. Ring )   &    |- F = ( R maMul <. M , N , O >. )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> O e. Fin )   &    |- .+ = ( +g ` R )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Y e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Z e. ( B ^m ( N X. O ) ) )   =>    |- ( ph -> ( ( X oF .+ Y ) F Z ) = ( ( X F Z ) oF .+ ( Y F Z ) ) )
Theorem	mamudir 20210	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 )   &    |- ( ph -> R e. Ring )   &    |- F = ( R maMul <. M , N , O >. )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> O e. Fin )   &    |- .+ = ( +g ` R )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Y e. ( B ^m ( N X. O ) ) )   &    |- ( ph -> Z e. ( B ^m ( N X. O ) ) )   =>    |- ( ph -> ( X F ( Y oF .+ Z ) ) = ( ( X F Y ) oF .+ ( X F Z ) ) )
Theorem	mamuvs1 20211	Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- B = ( Base ` R )   &    |- ( ph -> R e. Ring )   &    |- F = ( R maMul <. M , N , O >. )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> O e. Fin )   &    |- .x. = ( .r ` R )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Z e. ( B ^m ( N X. O ) ) )   =>    |- ( ph -> ( ( ( ( M X. N ) X. { X } ) oF .x. Y ) F Z ) = ( ( ( M X. O ) X. { X } ) oF .x. ( Y F Z ) ) )
Theorem	mamuvs2 20212	Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
|- ( ph -> R e. CRing )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- F = ( R maMul <. M , N , O >. )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> O e. Fin )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. ( B ^m ( N X. O ) ) )   =>    |- ( ph -> ( X F ( ( ( N X. O ) X. { Y } ) oF .x. Z ) ) = ( ( ( M X. O ) X. { Y } ) oF .x. ( X F Z ) ) )
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 20235. 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.
Syntax	cmat 20213	Syntax for the square matrix algebra.
class Mat
Definition	df-mat 20214*	Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
|- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
Theorem	matbas0pc 20215	There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
|- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
Theorem	matbas0 20216	There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
|- ( -. ( N e. Fin /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )
Theorem	matval 20217	Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
|- A = ( N Mat R )   &    |- G = ( R freeLMod ( N X. N ) )   &    |- .x. = ( R maMul <. N , N , N >. )   =>    |- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , .x. >. ) )
Theorem	matrcl 20218	Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   =>    |- ( X e. B -> ( N e. Fin /\ R e. _V ) )
Theorem	matbas 20219	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( Base ` G ) = ( Base ` A ) )
Theorem	matplusg 20220	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( +g ` G ) = ( +g ` A ) )
Theorem	matsca 20221	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> (Scalar ` G ) = (Scalar ` A ) )
Theorem	matvsca 20222	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( .s ` G ) = ( .s ` A ) )
Theorem	mat0 20223	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( 0g ` G ) = ( 0g ` A ) )
Theorem	matinvg 20224	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( invg ` G ) = ( invg ` A ) )
Theorem	mat0op 20225*	Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
|- A = ( N Mat R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` A ) = ( i e. N , j e. N |-> .0. ) )
Theorem	matsca2 20226	The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- A = ( N Mat R )   =>    |- ( ( N e. Fin /\ R e. V ) -> R = (Scalar ` A ) )
Theorem	matbas2 20227	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 e. Fin /\ R e. V ) -> ( K ^m ( N X. N ) ) = ( Base ` A ) )
Theorem	matbas2i 20228	A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
|- A = ( N Mat R )   &    |- K = ( Base ` R )   &    |- B = ( Base ` A )   =>    |- ( M e. B -> M e. ( K ^m ( N X. N ) ) )
Theorem	matbas2d 20229*	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 )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> R e. V )   &    |- ( ( ph /\ x e. N /\ y e. N ) -> C e. K )   =>    |- ( ph -> ( x e. N , y e. N |-> C ) e. B )
Theorem	eqmat 20230*	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 )   =>    |- ( ( X e. B /\ Y e. B ) -> ( X = Y <-> A. i e. N A. j e. N ( i X j ) = ( i Y j ) ) )
Theorem	matecl 20231	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 )   =>    |- ( ( I e. N /\ J e. N /\ M e. ( Base ` A ) ) -> ( I M J ) e. K )
Theorem	matecld 20232	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 )   &    |- ( ph -> I e. N )   &    |- ( ph -> J e. N )   &    |- ( ph -> M e. B )   =>    |- ( ph -> ( I M J ) e. K )
Theorem	matplusg2 20233	Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .+b = ( +g ` A )   &    |- .+ = ( +g ` R )   =>    |- ( ( X e. B /\ Y e. B ) -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	matvsca2 20234	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 )   &    |- .x. = ( .s ` A )   &    |- .X. = ( .r ` R )   &    |- C = ( N X. N )   =>    |- ( ( X e. K /\ Y e. B ) -> ( X .x. Y ) = ( ( C X. { X } ) oF .X. Y ) )
Theorem	matlmod 20235	The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
|- A = ( N Mat R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> A e. LMod )
Theorem	matgrp 20236	The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
|- A = ( N Mat R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> A e. Grp )
Theorem	matvscl 20237	Closure of the scalar multiplication in the matrix ring. (lmodvscl 18880 analog.) (Contributed by AV, 27-Nov-2019.)
|- K = ( Base ` R )   &    |- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .x. = ( .s ` A )   =>    |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( C e. K /\ X e. B ) ) -> ( C .x. X ) e. B )
Theorem	matsubg 20238	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 X. N ) )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( -g ` G ) = ( -g ` A ) )
Theorem	matplusgcell 20239	Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .+b = ( +g ` A )   &    |- .+ = ( +g ` R )   =>    |- ( ( ( X e. B /\ Y e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( X .+b Y ) J ) = ( ( I X J ) .+ ( I Y J ) ) )
Theorem	matsubgcell 20240	Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- S = ( -g ` A )   &    |- .- = ( -g ` R )   =>    |- ( ( R e. Ring /\ ( X e. B /\ Y e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( X S Y ) J ) = ( ( I X J ) .- ( I Y J ) ) )
Theorem	matinvgcell 20241	Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- V = ( invg ` R )   &    |- W = ( invg ` A )   =>    |- ( ( R e. Ring /\ X e. B /\ ( I e. N /\ J e. N ) ) -> ( I ( W ` X ) J ) = ( V ` ( I X J ) ) )
Theorem	matvscacell 20242	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 )   &    |- .x. = ( .s ` A )   &    |- .X. = ( .r ` R )   =>    |- ( ( R e. Ring /\ ( X e. K /\ Y e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( X .x. Y ) J ) = ( X .X. ( I Y J ) ) )
Theorem	matgsum 20243*	Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` A )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> J e. W )   &    |- ( ph -> R e. Ring )   &    |- ( ( ph /\ y e. J ) -> ( i e. N , j e. N |-> U ) e. B )   &    |- ( ph -> ( y e. J |-> ( i e. N , j e. N |-> U ) ) finSupp .0. )   =>    |- ( ph -> ( A gsumg ( y e. J |-> ( i e. N , j e. N |-> U ) ) ) = ( i e. N , j e. N |-> ( R gsumg ( y e. J |-> U ) ) ) )
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 20249) and an associative algebra (see matassa 20250). Additionally, theorems for the identity matrix and transposed matrices are provided.
Theorem	matmulr 20244	Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
|- A = ( N Mat R )   &    |- .x. = ( R maMul <. N , N , N >. )   =>    |- ( ( N e. Fin /\ R e. V ) -> .x. = ( .r ` A ) )
Theorem	mamumat1cl 20245*	The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
|- B = ( Base ` R )   &    |- ( ph -> R e. Ring )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )   &    |- ( ph -> M e. Fin )   =>    |- ( ph -> I e. ( B ^m ( M X. M ) ) )
Theorem	mat1comp 20246*	The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
|- B = ( Base ` R )   &    |- ( ph -> R e. Ring )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )   &    |- ( ph -> M e. Fin )   =>    |- ( ( A e. M /\ J e. M ) -> ( A I J ) = if ( A = J , .1. , .0. ) )
Theorem	mamulid 20247*	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 )   &    |- ( ph -> R e. Ring )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- F = ( R maMul <. M , M , N >. )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   =>    |- ( ph -> ( I F X ) = X )
Theorem	mamurid 20248*	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 )   &    |- ( ph -> R e. Ring )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- I = ( i e. M , j e. M |-> if ( i = j , .1. , .0. ) )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- F = ( R maMul <. N , M , M >. )   &    |- ( ph -> X e. ( B ^m ( N X. M ) ) )   =>    |- ( ph -> ( X F I ) = X )
Theorem	matring 20249	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 e. Fin /\ R e. Ring ) -> A e. Ring )
Theorem	matassa 20250	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 e. Fin /\ R e. CRing ) -> A e. AssAlg )
Theorem	matmulcell 20251*	Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .x. = ( .r ` R )   &    |- .X. = ( .r ` A )   =>    |- ( ( R e. Ring /\ ( X e. B /\ Y e. B ) /\ ( I e. N /\ J e. N ) ) -> ( I ( X .X. Y ) J ) = ( R gsumg ( j e. N |-> ( ( I X j ) ( .r ` R ) ( j Y J ) ) ) ) )
Theorem	mpt2matmul 20252*	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 )   &    |- .X. = ( .r ` A )   &    |- .x. = ( .r ` R )   &    |- ( ph -> R e. V )   &    |- ( ph -> N e. Fin )   &    |- X = ( i e. N , j e. N |-> C )   &    |- Y = ( i e. N , j e. N |-> E )   &    |- ( ( ph /\ i e. N /\ j e. N ) -> C e. B )   &    |- ( ( ph /\ i e. N /\ j e. N ) -> E e. B )   &    |- ( ( ph /\ ( k = i /\ m = j ) ) -> D = C )   &    |- ( ( ph /\ ( m = i /\ l = j ) ) -> F = E )   &    |- ( ( ph /\ k e. N /\ m e. N ) -> D e. U )   &    |- ( ( ph /\ m e. N /\ l e. N ) -> F e. W )   =>    |- ( ph -> ( X .X. Y ) = ( k e. N , l e. N |-> ( R gsumg ( m e. N |-> ( D .x. F ) ) ) ) )
Theorem	mat1 20253*	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. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) = ( i e. N , j e. N |-> if ( i = j , .1. , .0. ) ) )
Theorem	mat1ov 20254	Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
|- A = ( N Mat R )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> I e. N )   &    |- ( ph -> J e. N )   &    |- U = ( 1r ` A )   =>    |- ( ph -> ( I U J ) = if ( I = J , .1. , .0. ) )
Theorem	mat1bas 20255	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 e. Ring /\ N e. Fin ) -> .1. e. B )
Theorem	matsc 20256*	The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
|- A = ( N Mat R )   &    |- K = ( Base ` R )   &    |- .x. = ( .s ` A )   &    |- .0. = ( 0g ` R )   =>    |- ( ( N e. Fin /\ R e. Ring /\ L e. K ) -> ( L .x. ( 1r ` A ) ) = ( i e. N , j e. N |-> if ( i = j , L , .0. ) ) )
Theorem	ofco2 20257	Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
|- ( ( ( F e. _V /\ G e. _V ) /\ ( Fun H /\ ( F o. H ) e. _V /\ ( G o. H ) e. _V ) ) -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )
Theorem	oftpos 20258	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.)
|- ( ( F e. V /\ G e. W ) -> tpos ( F oF R G ) = (tpos F oF Rtpos G ) )
Theorem	mattposcl 20259	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 )   =>    |- ( M e. B -> tpos M e. B )
Theorem	mattpostpos 20260	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 )   =>    |- ( M e. B -> tpos tpos M = M )
Theorem	mattposvs 20261	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 )   &    |- .x. = ( .s ` A )   =>    |- ( ( X e. K /\ Y e. B ) -> tpos ( X .x. Y ) = ( X .x. tpos Y ) )
Theorem	mattpos1 20262	The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
|- A = ( N Mat R )   &    |- .1. = ( 1r ` A )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> tpos .1. = .1. )
Theorem	tposmap 20263	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 e. ( B ^m ( I X. J ) ) -> tpos A e. ( B ^m ( J X. I ) ) )
Theorem	mamutpos 20264	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 )   &    |- ( ph -> R e. CRing )   &    |- ( ph -> M e. Fin )   &    |- ( ph -> N e. Fin )   &    |- ( ph -> P e. Fin )   &    |- ( ph -> X e. ( B ^m ( M X. N ) ) )   &    |- ( ph -> Y e. ( B ^m ( N X. P ) ) )   =>    |- ( ph -> tpos ( X F Y ) = (tpos Y Gtpos X ) )
Theorem	mattposm 20265	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 )   &    |- .x. = ( .r ` A )   =>    |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> tpos ( X .x. Y ) = (tpos Y .x. tpos X ) )
Theorem	matgsumcl 20266*	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 e. CRing /\ M e. B ) -> ( U gsumg ( r e. N |-> ( r M r ) ) ) e. ( Base ` R ) )
Theorem	madetsumid 20267*	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 )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. CRing /\ M e. B /\ P = ( _I |` N ) ) -> ( ( ( Y o. S ) ` P ) .x. ( U gsumg ( r e. N |-> ( ( P ` r ) M r ) ) ) ) = ( U gsumg ( r e. N |-> ( r M r ) ) ) )
Theorem	matepmcl 20268*	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 e. Ring /\ Q e. P /\ M e. B ) -> A. n e. N ( ( Q ` n ) M n ) e. ( Base ` R ) )
Theorem	matepm2cl 20269*	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 e. Ring /\ Q e. P /\ M e. B ) -> A. n e. N ( n M ( Q ` n ) ) e. ( Base ` R ) )
Theorem	madetsmelbas 20270*	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 e. CRing /\ M e. B /\ Q e. P ) -> ( ( ( Y o. S ) ` Q ) ( .r ` R ) ( G gsumg ( n e. N |-> ( ( Q ` n ) M n ) ) ) ) e. ( Base ` R ) )
Theorem	madetsmelbas2 20271*	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 e. CRing /\ M e. B /\ Q e. P ) -> ( ( ( Y o. S ) ` Q ) ( .r ` R ) ( G gsumg ( n e. N |-> ( n M ( Q ` n ) ) ) ) ) e. ( Base ` R ) )
11.2.4  Matrices of dimension 0 and 1 As already mentioned before, and shown in mat0dimbas0 20272, 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 20276. 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 20293.
Theorem	mat0dimbas0 20272	The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
|- ( R e. V -> ( Base ` ( (/) Mat R ) ) = { (/) } )
Theorem	mat0dim0 20273	The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
|- A = ( (/) Mat R )   =>    |- ( R e. Ring -> ( 0g ` A ) = (/) )
Theorem	mat0dimid 20274	The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
|- A = ( (/) Mat R )   =>    |- ( R e. Ring -> ( 1r ` A ) = (/) )
Theorem	mat0dimscm 20275	The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
|- A = ( (/) Mat R )   =>    |- ( ( R e. Ring /\ X e. ( Base ` R ) ) -> ( X ( .s ` A ) (/) ) = (/) )
Theorem	mat0dimcrng 20276	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 e. Ring -> A e. CRing )
Theorem	mat1dimelbas 20277*	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 e. Ring /\ E e. V ) -> ( M e. ( Base ` A ) <-> E. r e. B M = { <. O , r >. } ) )
Theorem	mat1dimbas 20278	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 e. Ring /\ E e. V /\ X e. B ) -> { <. O , X >. } e. ( Base ` A ) )
Theorem	mat1dim0 20279	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 e. Ring /\ E e. V ) -> ( 0g ` A ) = { <. O , ( 0g ` R ) >. } )
Theorem	mat1dimid 20280	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 e. Ring /\ E e. V ) -> ( 1r ` A ) = { <. O , ( 1r ` R ) >. } )
Theorem	mat1dimscm 20281	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 e. Ring /\ E e. V ) /\ ( X e. B /\ Y e. B ) ) -> ( X ( .s ` A ) { <. O , Y >. } ) = { <. O , ( X ( .r ` R ) Y ) >. } )
Theorem	mat1dimmul 20282	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 e. Ring /\ E e. V ) /\ ( X e. B /\ Y e. B ) ) -> ( { <. O , X >. } ( .r ` A ) { <. O , Y >. } ) = { <. O , ( X ( .r ` R ) Y ) >. } )
Theorem	mat1dimcrng 20283	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 e. CRing /\ E e. V ) -> A e. CRing )
Theorem	mat1f1o 20284*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V ) -> F : K -1-1-onto-> B )
Theorem	mat1rhmval 20285*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) = { <. O , X >. } )
Theorem	mat1rhmelval 20286*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( E ( F ` X ) E ) = X )
Theorem	mat1rhmcl 20287*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) e. B )
Theorem	mat1f 20288*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V ) -> F : K --> B )
Theorem	mat1ghm 20289*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V ) -> F e. ( R GrpHom A ) )
Theorem	mat1mhm 20290*	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 = ( x e. K |-> { <. O , x >. } )   &    |- M = (mulGrp ` R )   &    |- N = (mulGrp ` A )   =>    |- ( ( R e. Ring /\ E e. V ) -> F e. ( M MndHom N ) )
Theorem	mat1rhm 20291*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V ) -> F e. ( R RingHom A ) )
Theorem	mat1rngiso 20292*	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 = ( x e. K |-> { <. O , x >. } )   =>    |- ( ( R e. Ring /\ E e. V ) -> F e. ( R RingIso A ) )
Theorem	mat1ric 20293	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 e. Ring /\ E e. V ) -> R ~=r 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 la .* I of the identity matrix I. Its effect on a vector is scalar multiplication by la [see scmatscm 20319!]". 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 20296 and df-scmat 20297), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20307), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20326), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20329) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20327).
Syntax	cdmat 20294	Extend class notation for the algebra of diagonal matrices.
class DMat
Syntax	cscmat 20295	Extend class notation for the algebra of scalar matrices.
class ScMat
Definition	df-dmat 20296*	Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
|- DMat = ( n e. Fin , r e. _V |-> { m e. ( Base ` ( n Mat r ) ) | A. i e. n A. j e. n ( i =/= j -> ( i m j ) = ( 0g ` r ) ) } )
Definition	df-scmat 20297*	Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
|- ScMat = ( n e. Fin , r e. _V |-> [_ ( n Mat r ) / a ]_ { m e. ( Base ` a ) | E. c e. ( Base ` r ) m = ( c ( .s ` a ) ( 1r ` a ) ) } )
Theorem	dmatval 20298*	The set of N x N diagonal matrices over (a ring) R. (Contributed by AV, 8-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( N e. Fin /\ R e. V ) -> D = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
Theorem	dmatel 20299*	A N x N diagonal matrix over (a ring) R. (Contributed by AV, 18-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( M e. D <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )
Theorem	dmatmat 20300	An N x N diagonal matrix over (the ring) R is an N x N matrix over (the ring) R. (Contributed by AV, 18-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( N e. Fin /\ R e. V ) -> ( M e. D -> M e. B ) )