:: Determinant of Some Matrices of Field Elements
:: by Yatsuka Nakamura
::
:: Received January 4, 2006
:: Copyright (c) 2006-2021 Association of Mizar Users

theorem Th1: :: MATRIX_7:1

for f being Permutation of (Seg 2) holds
( f = <*1,2*> or f = <*2,1*> )

proof end;

theorem Th2: :: MATRIX_7:2

for f being FinSequence st ( f = <*1,2*> or f = <*2,1*> ) holds
f is Permutation of (Seg 2)

proof end;

theorem Th3: :: MATRIX_7:3

Permutations 2 = {<*1,2*>,<*2,1*>}

proof end;

theorem Th4: :: MATRIX_7:4

for p being Permutation of (Seg 2) st p is being_transposition holds
p = <*2,1*>

proof end;

theorem Th5: :: MATRIX_7:5

for D being non empty set
for f being FinSequence of D
for k2 being Nat st 1 <= k2 & k2 < len f holds
f = (mid (f,1,k2)) ^ (mid (f,(k2 + 1),(len f)))

proof end;

theorem Th6: :: MATRIX_7:6

for D being non empty set
for f being FinSequence of D st 2 <= len f holds
f = (f | ((len f) -' 2)) ^ (mid (f,((len f) -' 1),(len f)))

proof end;

theorem Th7: :: MATRIX_7:7

for D being non empty set
for f being FinSequence of D st 1 <= len f holds
f = (f | ((len f) -' 1)) ^ (mid (f,(len f),(len f)))

proof end;

Lm1: <*1,2*> <> <*2,1*>
by FINSEQ_1:77;

theorem Th8: :: MATRIX_7:8

for a being Element of (Group_of_Perm 2) st ex q being Element of Permutations 2 st
( q = a & q is being_transposition ) holds
a = <*2,1*>

proof end;

theorem :: MATRIX_7:9

for n being Nat
for a, b being Element of (Group_of_Perm n)
for pa, pb being Element of Permutations n st a = pa & b = pb holds
a * b = pb * pa by MATRIX_1:def 13;

theorem Th10: :: MATRIX_7:10

for a, b being Element of (Group_of_Perm 2) st ex p being Element of Permutations 2 st
( p = a & p is being_transposition ) & ex q being Element of Permutations 2 st
( q = b & q is being_transposition ) holds
a * b = <*1,2*>

proof end;

theorem Th11: :: MATRIX_7:11

for l being FinSequence of (Group_of_Perm 2) st (len l) mod 2 = 0 & ( for i being Nat st i in dom l holds
ex q being Element of Permutations 2 st
( l . i = q & q is being_transposition ) ) holds
Product l = <*1,2*>

proof end;

theorem :: MATRIX_7:12

for K being Field
for M being Matrix of 2,K holds Det M = ((M * (1,1)) * (M * (2,2))) - ((M * (1,2)) * (M * (2,1)))

proof end;

definition

let n be Nat;
let K be commutative Ring;
let M be Matrix of n,K;
let a be Element of K;
:: original: *
redefine func a * M -> Matrix of n,K;
coherence
a * M is Matrix of n,K

proof end;

end;

theorem Th13: :: MATRIX_7:13

for K being Ring
for n, m being Nat holds
( len (0. (K,n,m)) = n & dom (0. (K,n,m)) = Seg n )

proof end;

theorem Th14: :: MATRIX_7:14

for K being Ring
for n being Nat
for p being Element of Permutations n
for i being Nat st i in Seg n holds
p . i in Seg n

proof end;

theorem :: MATRIX_7:15

for K being Ring
for n being Nat st n >= 1 holds
Det (0. (K,n,n)) = 0. K

proof end;

definition

let x be object ;
let y be set ;
let a, b be object ;

func IFIN (x,y,a,b) -> object equals :Def1: :: MATRIX_7:def 1
a if x in y
otherwise b;

correctness
coherence
( ( x in y implies a is object ) & ( not x in y implies b is object ) );
consistency
for b₁ being object holds verum;
;

end;

:: deftheorem Def1 defines IFIN MATRIX_7:def 1 :
for x being object
for y being set
for a, b being object holds
( ( x in y implies IFIN (x,y,a,b) = a ) & ( not x in y implies IFIN (x,y,a,b) = b ) );

theorem :: MATRIX_7:16

for K being Ring
for n being Nat st n >= 1 holds
Det (1. (K,n)) = 1_ K

proof end;

definition

let n be Nat;
let K be Field;
let M be Matrix of n,K;

:: original: diagonal
redefine attr M is diagonal means :: MATRIX_7:def 2
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
M * (i,j) = 0. K;

compatibility
( M is diagonal iff for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
M * (i,j) = 0. K )

proof end;

end;

:: deftheorem defines diagonal MATRIX_7:def 2 :
for n being Nat
for K being Field
for M being Matrix of n,K holds
( M is diagonal iff for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
M * (i,j) = 0. K );

theorem :: MATRIX_7:17

for K being Field
for n being Nat
for A being Matrix of n,K st n >= 1 & A is V196(b₁) holds
Det A = the multF of K $$ (diagonal_of_Matrix A)

proof end;

theorem Th18: :: MATRIX_7:18

for n being Nat
for p being Element of Permutations n holds p " is Element of Permutations n

proof end;

definition

let n be Nat;
let p be Element of Permutations n;
:: original: "
redefine func p " -> Element of Permutations n;
coherence
p " is Element of Permutations n by Th18;

end;

theorem :: MATRIX_7:19

canceled;

::$CT

theorem :: MATRIX_7:20

for G being Group
for f1, f2 being FinSequence of G holds (Product (f1 ^ f2)) " = ((Product f2) ") * ((Product f1) ")

proof end;

definition

let G be Group;
let f be FinSequence of G;

::$CD

func f " -> FinSequence of G means :Def3: :: MATRIX_7:def 4
( len it = len f & ( for i being Nat st i in dom f holds
it /. i = (f /. i) " ) );

existence
ex b₁ being FinSequence of G st
( len b₁ = len f & ( for i being Nat st i in dom f holds
b₁ /. i = (f /. i) " ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of G st len b₁ = len f & ( for i being Nat st i in dom f holds
b₁ /. i = (f /. i) " ) & len b₂ = len f & ( for i being Nat st i in dom f holds
b₂ /. i = (f /. i) " ) holds
b₁ = b₂

proof end;

end;

:: deftheorem MATRIX_7:def 3 :
canceled;

:: deftheorem Def3 defines " MATRIX_7:def 4 :
for G being Group
for f, b₃ being FinSequence of G holds
( b₃ = f " iff ( len b₃ = len f & ( for i being Nat st i in dom f holds
b₃ /. i = (f /. i) " ) ) );

theorem Th20: :: MATRIX_7:21

for G being Group holds (<*> the carrier of G) " = <*> the carrier of G

proof end;

theorem Th21: :: MATRIX_7:22

for G being Group
for f, g being FinSequence of G holds (f ^ g) " = (f ") ^ (g ")

proof end;

theorem Th22: :: MATRIX_7:23

for G being Group
for a being Element of G holds <*a*> " = <*(a ")*>

proof end;

theorem Th23: :: MATRIX_7:24

for G being Group
for f being FinSequence of G holds Product (f ^ ((Rev f) ")) = 1_ G

proof end;

theorem Th24: :: MATRIX_7:25

for G being Group
for f being FinSequence of G holds Product (((Rev f) ") ^ f) = 1_ G

proof end;

theorem Th25: :: MATRIX_7:26

for G being Group
for f being FinSequence of G holds (Product f) " = Product ((Rev f) ")

proof end;

theorem Th26: :: MATRIX_7:27

for n being Nat
for ITP being Element of Permutations n
for ITG being Element of (Group_of_Perm n) st ITG = ITP & n >= 1 holds
ITP " = ITG "

proof end;

Lm2: for n being Nat
for IT being Element of Permutations n st IT is even & n >= 1 holds
IT " is even

proof end;

theorem Th27: :: MATRIX_7:28

for n being Nat
for IT being Element of Permutations n st n >= 1 holds
( IT is even iff IT " is even )

proof end;

theorem Th28: :: MATRIX_7:29

for n being Nat
for K being commutative Ring
for p being Element of Permutations n
for x being Element of K st n >= 1 holds
- (x,p) = - (x,(p "))

proof end;

theorem :: MATRIX_7:30

for K being commutative Ring
for f1, f2 being FinSequence of K holds the multF of K $$ (f1 ^ f2) = ( the multF of K $$ f1) * ( the multF of K $$ f2) by FINSOP_1:5;

theorem Th30: :: MATRIX_7:31

for K being commutative Ring
for R1, R2 being FinSequence of K st R1,R2 are_fiberwise_equipotent holds
the multF of K $$ R1 = the multF of K $$ R2

proof end;

theorem Th31: :: MATRIX_7:32

for n being Nat
for K being commutative Ring
for p being Element of Permutations n
for f, g being FinSequence of K st len f = n & g = f * p holds
f,g are_fiberwise_equipotent

proof end;

theorem :: MATRIX_7:33

for n being Nat
for K being commutative Ring
for p being Element of Permutations n
for f, g being FinSequence of K st n >= 1 & len f = n & g = f * p holds
the multF of K $$ f = the multF of K $$ g by Th30, Th31;

theorem Th33: :: MATRIX_7:34

for n being Nat
for K being Ring
for p being Element of Permutations n
for f being FinSequence of K st n >= 1 & len f = n holds
f * p is FinSequence of K

proof end;

theorem Th34: :: MATRIX_7:35

for n being Nat
for K being commutative Ring
for p being Element of Permutations n
for A being Matrix of n,K st n >= 1 holds
Path_matrix ((p "),(A @)) = (Path_matrix (p,A)) * (p ")

proof end;

theorem Th35: :: MATRIX_7:36

for n being Nat
for K being commutative Ring
for p being Element of Permutations n
for A being Matrix of n,K st n >= 1 holds
(Path_product (A @)) . (p ") = (Path_product A) . p

proof end;

theorem :: MATRIX_7:37

for n being Nat
for K being commutative Ring
for A being Matrix of n,K st n >= 1 holds
Det A = Det (A @)

proof end;