P. 196 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gsumcom3 19501*	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 19502*	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 19503	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 19504	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 19505	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 19506	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 19507	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 19508	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 19531. 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 19509	Syntax for the square matrix algebra.
class Mat
Definition	df-mat 19510*	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 19511	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 19512	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 19513	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 19514	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 19515	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 19516	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 19517	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 19518	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 19519	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 19520	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 19521*	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 19522	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 19523	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 19524	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 19525*	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 19526*	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 19527	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 19528	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 19529	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 19530	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 19531	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 19532	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 19533	Closure of the scalar multiplication in the matrix ring. (lmodvscl 18186 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 19534	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 19535	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 19536	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 19537	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 19538	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 19539*	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 19545) and an associative algebra (see matassa 19546). Additionally, theorems for the identity matrix and transposed matrices are provided.
Theorem	matmulr 19540	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 19541*	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 19542*	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 19543*	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 19544*	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 19545	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 19546	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 19547*	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 19548*	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 19549*	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 19550	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 19551	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 19552*	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 19553	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 19554	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 19555	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 19556	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 19557	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 19558	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 19559	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 19560	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 19561	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 19562*	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 19563*	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 19564*	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 19565*	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 19566*	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 19567*	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 19568, 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 19572. 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 19589.
Theorem	mat0dimbas0 19568	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 19569	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 19570	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 19571	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 19572	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 19573*	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 19574	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 19575	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 19576	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 19577	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 19578	The ring 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 ) ) -> ( { <. O , X >. } ( .r ` A ) { <. O , Y >. } ) = { <. O , ( X ( .r ` R ) Y ) >. } )
Theorem	mat1dimcrng 19579	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 19580*	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 19581*	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 19582*	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 19583*	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 19584*	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 19585*	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 19586*	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 19587*	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 19588*	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 19589	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 19615!]". 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 19592 and df-scmat 19593), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 19603), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 19622), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 19625) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 19623).
Syntax	cdmat 19590	Extend class notation for the algebra of diagonal matrices.
class DMat
Syntax	cscmat 19591	Extend class notation for the algebra of scalar matrices.
class ScMat
Definition	df-dmat 19592*	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 19593*	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 19594*	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 19595*	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 19596	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 ) )
Theorem	dmatid 19597	The identity matrix is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( N e. Fin /\ R e. Ring ) -> ( 1r ` A ) e. D )
Theorem	dmatelnd 19598	An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( ( N e. Fin /\ R e. Ring /\ X e. D ) /\ ( I e. N /\ J e. N /\ I =/= J ) ) -> ( I X J ) = .0. )
Theorem	dmatmul 19599*	The product of two diagonal matrices. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( X e. D /\ Y e. D ) ) -> ( X ( .r ` A ) Y ) = ( x e. N , y e. N |-> if ( x = y , ( ( x X y ) ( .r ` R ) ( x Y y ) ) , .0. ) ) )
Theorem	dmatsubcl 19600	The difference of two diagonal matrices is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
|- A = ( N Mat R )   &    |- B = ( Base ` A )   &    |- .0. = ( 0g ` R )   &    |- D = ( N DMat R )   =>    |- ( ( ( N e. Fin /\ R e. Ring ) /\ ( X e. D /\ Y e. D ) ) -> ( X ( -g ` A ) Y ) e. D )