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Theorem chcoeffeq 20691
Description: The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
chcoeffeq.a |- A = ( N Mat R )
chcoeffeq.b |- B = ( Base ` A )
chcoeffeq.p |- P = (Poly1 ` R )
chcoeffeq.y |- Y = ( N Mat P )
chcoeffeq.r |- .X. = ( .r ` Y )
chcoeffeq.s |- .- = ( -g ` Y )
chcoeffeq.0 |- .0. = ( 0g ` Y )
chcoeffeq.t |- T = ( N matToPolyMat R )
chcoeffeq.c |- C = ( N CharPlyMat R )
chcoeffeq.k |- K = ( C ` M )
chcoeffeq.g |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )
chcoeffeq.w |- W = ( Base ` Y )
chcoeffeq.1 |- .1. = ( 1r ` A )
chcoeffeq.m |- .* = ( .s ` A )
chcoeffeq.u |- U = ( N cPolyMatToMat R )
Assertion
Ref Expression
chcoeffeq |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) )
Distinct variable groups:   A, n   B, n   n, G   n, K   n, M   n, N   R, n   U, n   n, Y   .1. , n   .* , n   n, b, s, A   B, b, s   M, b, s   N, b, s   P, b, n, s   R, b, s   T, b, n, s   n, W   Y, b, s   .0. , n   .X. , n   .- , b, n, s
Allowed substitution hints:   C( n, s, b)   .X. ( s, b)   U( s, b)   .1. ( s, b)   G( s, b)   .* ( s, b)   K( s, b)   W( s, b)   .0. ( s, b)
Proof of Theorem chcoeffeq
StepHypRef Expression
1 chcoeffeq.a . . 3 |- A = ( N Mat R )
2 chcoeffeq.b . . 3 |- B = ( Base ` A )
3 chcoeffeq.p . . 3 |- P = (Poly1 ` R )
4 chcoeffeq.y . . 3 |- Y = ( N Mat P )
5 chcoeffeq.t . . 3 |- T = ( N matToPolyMat R )
6 chcoeffeq.r . . 3 |- .X. = ( .r ` Y )
7 chcoeffeq.s . . 3 |- .- = ( -g ` Y )
8 chcoeffeq.0 . . 3 |- .0. = ( 0g ` Y )
9 chcoeffeq.g . . 3 |- G = ( n e. NN0 |-> if ( n = 0 , ( .0. .- ( ( T ` M ) .X. ( T ` ( b ` 0 ) ) ) ) , if ( n = ( s + 1 ) , ( T ` ( b ` s ) ) , if ( ( s + 1 ) < n , .0. , ( ( T ` ( b ` ( n - 1 ) ) ) .- ( ( T ` M ) .X. ( T ` ( b ` n ) ) ) ) ) ) ) )
10 eqid 2622 . . 3 |- ( N ConstPolyMat R ) = ( N ConstPolyMat R )
11 eqid 2622 . . 3 |- ( .s ` Y ) = ( .s ` Y )
12 eqid 2622 . . 3 |- ( 1r ` Y ) = ( 1r ` Y )
13 eqid 2622 . . 3 |- (var1 ` R ) = (var1 ` R )
14 eqid 2622 . . 3 |- ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) = ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) )
15 eqid 2622 . . 3 |- ( N maAdju P ) = ( N maAdju P )
16 chcoeffeq.w . . 3 |- W = ( Base ` Y )
17 eqid 2622 . . 3 |- (Poly1 ` A ) = (Poly1 ` A )
18 eqid 2622 . . 3 |- (var1 ` A ) = (var1 ` A )
19 eqid 2622 . . 3 |- ( .s ` (Poly1 ` A ) ) = ( .s ` (Poly1 ` A ) )
20 eqid 2622 . . 3 |- (.g ` (mulGrp ` (Poly1 ` A ) ) ) = (.g ` (mulGrp ` (Poly1 ` A ) ) )
21 chcoeffeq.u . . 3 |- U = ( N cPolyMatToMat R )
22 eqid 2622 . . 3 |- ( N pMatToMatPoly R ) = ( N pMatToMatPoly R )
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22cpmadumatpoly 20688 . 2 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
24 eqid 2622 . . . . . . 7 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
25 eqid 2622 . . . . . . 7 |- (algSc ` P ) = (algSc ` P )
26 chcoeffeq.c . . . . . . 7 |- C = ( N CharPlyMat R )
27 chcoeffeq.k . . . . . . 7 |- K = ( C ` M )
28 eqid 2622 . . . . . . 7 |- ( K ( .s ` Y ) ( 1r ` Y ) ) = ( K ( .s ` Y ) ( 1r ` Y ) )
29 chcoeffeq.1 . . . . . . 7 |- .1. = ( 1r ` A )
30 chcoeffeq.m . . . . . . 7 |- .* = ( .s ` A )
311, 2, 3, 4, 13, 24, 11, 12, 25, 26, 27, 28, 29, 30, 5, 16, 17, 18, 19, 20, 22cpmidpmat 20678 . . . . . 6 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
32 eqid 2622 . . . . . . . 8 |- ( N CharPlyMat R ) = ( N CharPlyMat R )
331, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6cpmadurid 20672 . . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) = ( ( ( N CharPlyMat R ) ` M ) ( .s ` Y ) ( 1r ` Y ) ) )
3426fveq1i 6192 . . . . . . . . . . 11 |- ( C ` M ) = ( ( N CharPlyMat R ) ` M )
3527, 34eqtri 2644 . . . . . . . . . 10 |- K = ( ( N CharPlyMat R ) ` M )
3635a1i 11 . . . . . . . . 9 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> K = ( ( N CharPlyMat R ) ` M ) )
3736eqcomd 2628 . . . . . . . 8 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( N CharPlyMat R ) ` M ) = K )
3837oveq1d 6665 . . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( ( N CharPlyMat R ) ` M ) ( .s ` Y ) ( 1r ` Y ) ) = ( K ( .s ` Y ) ( 1r ` Y ) ) )
3933, 38eqtrd 2656 . . . . . 6 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) = ( K ( .s ` Y ) ( 1r ` Y ) ) )
40 fveq2 6191 . . . . . . . . 9 |- ( ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) = ( K ( .s ` Y ) ( 1r ` Y ) ) -> ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) )
41 simpr 477 . . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
4241adantr 481 . . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
43 simpr 477 . . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) )
4442, 43eqeq12d 2637 . . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) <-> ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) )
451, 2, 3, 4, 6, 7, 8, 5, 26, 27, 9, 16, 29, 30, 21chcoeffeqlem 20690 . . . . . . . . . . . . . 14 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
4645adantr 481 . . . . . . . . . . . . 13 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
4746adantr 481 . . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
4844, 47sylbid 230 . . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) /\ ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
4948exp31 630 . . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) ) ) )
5049com24 95 . . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) ) ) )
5140, 50syl5 34 . . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) ) -> ( ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) = ( K ( .s ` Y ) ( 1r ` Y ) ) -> ( ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) ) ) )
5251ex 450 . . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) = ( K ( .s ` Y ) ( 1r ` Y ) ) -> ( ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) ) ) ) )
5352com24 95 . . . . . 6 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( ( N pMatToMatPoly R ) ` ( K ( .s ` Y ) ( 1r ` Y ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( ( (coe1 ` K ) ` n ) .* .1. ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> ( ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) = ( K ( .s ` Y ) ( 1r ` Y ) ) -> ( ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) ) ) ) )
5431, 39, 53mp2d 49 . . . . 5 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( ( s e. NN /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) ) )
5554impl 650 . . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ s e. NN ) /\ b e. ( B ^m ( 0 ... s ) ) ) -> ( ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
5655reximdva 3017 . . 3 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ s e. NN ) -> ( E. b e. ( B ^m ( 0 ... s ) ) ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> E. b e. ( B ^m ( 0 ... s ) ) A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
5756reximdva 3017 . 2 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) ( ( N pMatToMatPoly R ) ` ( ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) .X. ( ( N maAdju P ) ` ( ( (var1 ` R ) ( .s ` Y ) ( 1r ` Y ) ) .- ( T ` M ) ) ) ) ) = ( (Poly1 ` A ) gsumg ( n e. NN0 |-> ( ( U ` ( G ` n ) ) ( .s ` (Poly1 ` A ) ) ( n (.g ` (mulGrp ` (Poly1 ` A ) ) ) (var1 ` A ) ) ) ) ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) ) )
5823, 57mpd 15 1 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> E. s e. NN E. b e. ( B ^m ( 0 ... s ) ) A. n e. NN0 ( U ` ( G ` n ) ) = ( ( (coe1 ` K ) ` n ) .* .1. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942   .scvsca 15945   0gc0g 16100   gsumg cgsu 16101   -gcsg 17424  .gcmg 17540  mulGrpcmgp 18489   1rcur 18501   CRingccrg 18548  algSccascl 19311  var1cv1 19546  Poly1cpl1 19547  coe1cco1 19548   Mat cmat 20213   maAdju cmadu 20438   ConstPolyMat ccpmat 20508   matToPolyMat cmat2pmat 20509   cPolyMatToMat ccpmat2mat 20510   pMatToMatPoly cpm2mp 20597   CharPlyMat cchpmat 20631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-cur 7393  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-symg 17798  df-pmtr 17862  df-psgn 17911  df-evpm 17912  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-assa 19312  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mdet 20391  df-madu 20440  df-cpmat 20511  df-mat2pmat 20512  df-cpmat2mat 20513  df-decpmat 20568  df-pm2mp 20598  df-chpmat 20632
This theorem is referenced by:  cayhamlem3  20692
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