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Theorem pmatcollpw3lem 19807
Description: Lemma for pmatcollpw3 19808 and pmatcollpw3fi 19809: Write a polynomial matrix (over a commutative ring) as sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 8-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpw.p  |-  P  =  (Poly1 `  R )
pmatcollpw.c  |-  C  =  ( N Mat  P )
pmatcollpw.b  |-  B  =  ( Base `  C
)
pmatcollpw.m  |-  .*  =  ( .s `  C )
pmatcollpw.e  |-  .^  =  (.g
`  (mulGrp `  P )
)
pmatcollpw.x  |-  X  =  (var1 `  R )
pmatcollpw.t  |-  T  =  ( N matToPolyMat  R )
pmatcollpw3.a  |-  A  =  ( N Mat  R )
pmatcollpw3.d  |-  D  =  ( Base `  A
)
Assertion
Ref Expression
pmatcollpw3lem  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( M  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( M decompPMat  n ) ) ) ) )  ->  E. f  e.  ( D  ^m  I
) M  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( f `  n ) ) ) ) ) ) )
Distinct variable groups:    B, n    n, M    n, N    P, n    R, n    n, X    .^ , n    C, n    B, f    C, f, n    D, f   
f, I, n    f, M    f, N    R, f    T, f    f, X    .^ , f    .* , f
Allowed substitution hints:    A( f, n)    D( n)    P( f)    T( n)    .* ( n)

Proof of Theorem pmatcollpw3lem
Dummy variables  i 
j  k  l  x  y  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 5035 . . . . . . . . 9  |-  ( x  =  y  ->  dom  x  =  dom  y )
21dmeqd 5037 . . . . . . . 8  |-  ( x  =  y  ->  dom  dom  x  =  dom  dom  y )
3 oveq 6296 . . . . . . . . . 10  |-  ( x  =  y  ->  (
i x j )  =  ( i y j ) )
43fveq2d 5869 . . . . . . . . 9  |-  ( x  =  y  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i y j ) ) )
54fveq1d 5867 . . . . . . . 8  |-  ( x  =  y  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i y j ) ) `  k ) )
62, 2, 5mpt2eq123dv 6353 . . . . . . 7  |-  ( x  =  y  ->  (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  =  ( i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) ) )
7 fveq2 5865 . . . . . . . 8  |-  ( k  =  l  ->  (
(coe1 `  ( i y j ) ) `  k )  =  ( (coe1 `  ( i y j ) ) `  l ) )
87mpt2eq3dv 6357 . . . . . . 7  |-  ( k  =  l  ->  (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) )  =  ( i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) ) )
96, 8cbvmpt2v 6371 . . . . . 6  |-  ( x  e.  B ,  k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )  =  ( y  e.  B , 
l  e.  I  |->  ( i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) ) )
10 dmexg 6724 . . . . . . . . . . 11  |-  ( y  e.  B  ->  dom  y  e.  _V )
11 dmexg 6724 . . . . . . . . . . 11  |-  ( dom  y  e.  _V  ->  dom 
dom  y  e.  _V )
1210, 11syl 17 . . . . . . . . . 10  |-  ( y  e.  B  ->  dom  dom  y  e.  _V )
1312, 12jca 535 . . . . . . . . 9  |-  ( y  e.  B  ->  ( dom  dom  y  e.  _V  /\ 
dom  dom  y  e.  _V ) )
1413ad2antrl 734 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  (
y  e.  B  /\  l  e.  I )
)  ->  ( dom  dom  y  e.  _V  /\  dom  dom  y  e.  _V ) )
15 mpt2exga 6869 . . . . . . . 8  |-  ( ( dom  dom  y  e.  _V  /\  dom  dom  y  e.  _V )  ->  (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) )  e.  _V )
1614, 15syl 17 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  (
y  e.  B  /\  l  e.  I )
)  ->  ( i  e.  dom  dom  y , 
j  e.  dom  dom  y  |->  ( (coe1 `  (
i y j ) ) `  l ) )  e.  _V )
1716ralrimivva 2809 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  A. y  e.  B  A. l  e.  I 
( i  e.  dom  dom  y ,  j  e. 
dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) )  e.  _V )
18 simprr 766 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  I  =/=  (/) )
19 nn0ex 10875 . . . . . . . 8  |-  NN0  e.  _V
2019ssex 4547 . . . . . . 7  |-  ( I 
C_  NN0  ->  I  e. 
_V )
2120ad2antrl 734 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  I  e.  _V )
22 simp3 1010 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  M  e.  B )
2322adantr 467 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  M  e.  B )
249, 17, 18, 21, 23mpt2curryvald 7017 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
(curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M )  =  ( l  e.  I  |-> 
[_ M  /  y ]_ ( i  e.  dom  dom  y ,  j  e. 
dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) ) ) )
25 fveq2 5865 . . . . . . . . 9  |-  ( l  =  k  ->  (
(coe1 `  ( i y j ) ) `  l )  =  ( (coe1 `  ( i y j ) ) `  k ) )
2625mpt2eq3dv 6357 . . . . . . . 8  |-  ( l  =  k  ->  (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) )  =  ( i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) ) )
2726csbeq2dv 3781 . . . . . . 7  |-  ( l  =  k  ->  [_ M  /  y ]_ (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) )  =  [_ M  /  y ]_ (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) ) )
28 eqcom 2458 . . . . . . . . 9  |-  ( x  =  y  <->  y  =  x )
29 eqcom 2458 . . . . . . . . 9  |-  ( ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  =  ( i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) )  <->  ( i  e.  dom  dom  y , 
j  e.  dom  dom  y  |->  ( (coe1 `  (
i y j ) ) `  k ) )  =  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )
306, 28, 293imtr3i 269 . . . . . . . 8  |-  ( y  =  x  ->  (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) )  =  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )
3130cbvcsbv 3369 . . . . . . 7  |-  [_ M  /  y ]_ (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  k
) )  =  [_ M  /  x ]_ (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )
3227, 31syl6eq 2501 . . . . . 6  |-  ( l  =  k  ->  [_ M  /  y ]_ (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) )  =  [_ M  /  x ]_ (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )
3332cbvmptv 4495 . . . . 5  |-  ( l  e.  I  |->  [_ M  /  y ]_ (
i  e.  dom  dom  y ,  j  e.  dom  dom  y  |->  ( (coe1 `  ( i y j ) ) `  l
) ) )  =  ( k  e.  I  |-> 
[_ M  /  x ]_ ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )
3424, 33syl6eq 2501 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
(curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M )  =  ( k  e.  I  |-> 
[_ M  /  x ]_ ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) )
35 dmeq 5035 . . . . . . . . . . 11  |-  ( x  =  M  ->  dom  x  =  dom  M )
3635dmeqd 5037 . . . . . . . . . 10  |-  ( x  =  M  ->  dom  dom  x  =  dom  dom  M )
37 oveq 6296 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
i x j )  =  ( i M j ) )
3837fveq2d 5869 . . . . . . . . . . 11  |-  ( x  =  M  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i M j ) ) )
3938fveq1d 5867 . . . . . . . . . 10  |-  ( x  =  M  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
4036, 36, 39mpt2eq123dv 6353 . . . . . . . . 9  |-  ( x  =  M  ->  (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  =  ( i  e.  dom  dom  M ,  j  e.  dom  dom 
M  |->  ( (coe1 `  (
i M j ) ) `  k ) ) )
4140adantl 468 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  x  =  M )  ->  (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  =  ( i  e.  dom  dom  M ,  j  e.  dom  dom 
M  |->  ( (coe1 `  (
i M j ) ) `  k ) ) )
4222, 41csbied 3390 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  [_ M  /  x ]_ ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) )  =  ( i  e.  dom  dom  M ,  j  e.  dom  dom 
M  |->  ( (coe1 `  (
i M j ) ) `  k ) ) )
43 pmatcollpw.c . . . . . . . . . . . . 13  |-  C  =  ( N Mat  P )
44 eqid 2451 . . . . . . . . . . . . 13  |-  ( Base `  P )  =  (
Base `  P )
45 pmatcollpw.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  C
)
4643, 44, 45matbas2i 19447 . . . . . . . . . . . 12  |-  ( M  e.  B  ->  M  e.  ( ( Base `  P
)  ^m  ( N  X.  N ) ) )
47 elmapi 7493 . . . . . . . . . . . 12  |-  ( M  e.  ( ( Base `  P )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  P
) )
48 fdm 5733 . . . . . . . . . . . . . 14  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  dom  M  =  ( N  X.  N ) )
4948dmeqd 5037 . . . . . . . . . . . . 13  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  dom  dom 
M  =  dom  ( N  X.  N ) )
50 dmxpid 5054 . . . . . . . . . . . . 13  |-  dom  ( N  X.  N )  =  N
5149, 50syl6req 2502 . . . . . . . . . . . 12  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  N  =  dom  dom  M )
5246, 47, 513syl 18 . . . . . . . . . . 11  |-  ( M  e.  B  ->  N  =  dom  dom  M )
53523ad2ant3 1031 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  N  =  dom  dom  M )
5453adantr 467 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  N  =  dom  dom  M )
55 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  m  =  M )
5655oveqd 6307 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i m j )  =  ( i M j ) )
5756fveq2d 5869 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
5857fveq1d 5867 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
5954, 54, 58mpt2eq123dv 6353 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  dom  dom 
M ,  j  e. 
dom  dom  M  |->  ( (coe1 `  ( i M j ) ) `  k
) ) )
6022, 59csbied 3390 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) )  =  ( i  e.  dom  dom  M ,  j  e.  dom  dom 
M  |->  ( (coe1 `  (
i M j ) ) `  k ) ) )
6142, 60eqtr4d 2488 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  [_ M  /  x ]_ ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) )  =  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) )
6261adantr 467 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  [_ M  /  x ]_ ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  =  [_ M  /  m ]_ (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
6362mpteq2dv 4490 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( k  e.  I  |-> 
[_ M  /  x ]_ ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )  =  ( k  e.  I  |-> 
[_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) ) )
6434, 63eqtrd 2485 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
(curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M )  =  ( k  e.  I  |-> 
[_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) ) )
65 oveq 6296 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
6665adantl 468 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i m j )  =  ( i M j ) )
6766fveq2d 5869 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
6867fveq1d 5867 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
6968mpt2eq3dv 6357 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  k ) ) )
7022, 69csbied 3390 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) )  =  ( i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  k ) ) )
7170ad2antrr 732 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) )  =  ( i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  k ) ) )
72 pmatcollpw3.a . . . . . . 7  |-  A  =  ( N Mat  R )
73 eqid 2451 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
74 pmatcollpw3.d . . . . . . 7  |-  D  =  ( Base `  A
)
75 simpll1 1047 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  N  e.  Fin )
76 simpll2 1048 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  R  e.  CRing )
77 simp2 1009 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  /\  i  e.  N  /\  j  e.  N
)  ->  i  e.  N )
78 simp3 1010 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  /\  i  e.  N  /\  j  e.  N
)  ->  j  e.  N )
7923adantr 467 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  M  e.  B )
80793ad2ant1 1029 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  /\  i  e.  N  /\  j  e.  N
)  ->  M  e.  B )
8143, 44, 45, 77, 78, 80matecld 19451 . . . . . . . 8  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  /\  i  e.  N  /\  j  e.  N
)  ->  ( i M j )  e.  ( Base `  P
) )
82 ssel 3426 . . . . . . . . . . 11  |-  ( I 
C_  NN0  ->  ( k  e.  I  ->  k  e.  NN0 ) )
8382ad2antrl 734 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( k  e.  I  ->  k  e.  NN0 )
)
8483imp 431 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  k  e.  NN0 )
85843ad2ant1 1029 . . . . . . . 8  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  /\  i  e.  N  /\  j  e.  N
)  ->  k  e.  NN0 )
86 eqid 2451 . . . . . . . . 9  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
87 pmatcollpw.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
8886, 44, 87, 73coe1fvalcl 18805 . . . . . . . 8  |-  ( ( ( i M j )  e.  ( Base `  P )  /\  k  e.  NN0 )  ->  (
(coe1 `  ( i M j ) ) `  k )  e.  (
Base `  R )
)
8981, 85, 88syl2anc 667 . . . . . . 7  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  /\  i  e.  N  /\  j  e.  N
)  ->  ( (coe1 `  ( i M j ) ) `  k
)  e.  ( Base `  R ) )
9072, 73, 74, 75, 76, 89matbas2d 19448 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  k ) )  e.  D )
9171, 90eqeltrd 2529 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) )  e.  D
)
92 eqid 2451 . . . . 5  |-  ( k  e.  I  |->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) )  =  ( k  e.  I  |-> 
[_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
9391, 92fmptd 6046 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( k  e.  I  |-> 
[_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) ) : I --> D )
94 fvex 5875 . . . . . . 7  |-  ( Base `  A )  e.  _V
9574, 94eqeltri 2525 . . . . . 6  |-  D  e. 
_V
9695a1i 11 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  D  e.  _V )
9720adantr 467 . . . . 5  |-  ( ( I  C_  NN0  /\  I  =/=  (/) )  ->  I  e.  _V )
98 elmapg 7485 . . . . 5  |-  ( ( D  e.  _V  /\  I  e.  _V )  ->  ( ( k  e.  I  |->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )  e.  ( D  ^m  I )  <->  ( k  e.  I  |->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) ) : I --> D ) )
9996, 97, 98syl2an 480 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( ( k  e.  I  |->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )  e.  ( D  ^m  I )  <->  ( k  e.  I  |->  [_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) ) : I --> D ) )
10093, 99mpbird 236 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( k  e.  I  |-> 
[_ M  /  m ]_ ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )  e.  ( D  ^m  I ) )
10164, 100eqeltrd 2529 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
(curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M )  e.  ( D  ^m  I
) )
102 fveq1 5864 . . . . . . . . . . 11  |-  ( f  =  (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) ) `  M
)  ->  ( f `  n )  =  ( (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) `  n ) )
103102adantl 468 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) )  ->  ( f `  n )  =  ( (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) `  n ) )
104103adantr 467 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( f `  n
)  =  ( (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) `  n
) )
105 eqid 2451 . . . . . . . . . . . 12  |-  ( x  e.  B ,  k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )  =  ( x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )
106 dmexg 6724 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  B  ->  dom  x  e.  _V )
107 dmexg 6724 . . . . . . . . . . . . . . . . 17  |-  ( dom  x  e.  _V  ->  dom 
dom  x  e.  _V )
108106, 107syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  B  ->  dom  dom  x  e.  _V )
109108, 108jca 535 . . . . . . . . . . . . . . 15  |-  ( x  e.  B  ->  ( dom  dom  x  e.  _V  /\ 
dom  dom  x  e.  _V ) )
110109ad2antrl 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  /\  ( x  e.  B  /\  k  e.  I
) )  ->  ( dom  dom  x  e.  _V  /\ 
dom  dom  x  e.  _V ) )
111 mpt2exga 6869 . . . . . . . . . . . . . 14  |-  ( ( dom  dom  x  e.  _V  /\  dom  dom  x  e.  _V )  ->  (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  e.  _V )
112110, 111syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  /\  ( x  e.  B  /\  k  e.  I
) )  ->  (
i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) )  e.  _V )
113112ralrimivva 2809 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  A. x  e.  B  A. k  e.  I  ( i  e.  dom  dom  x , 
j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) )  e.  _V )
11421adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  I  e.  _V )
11523adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  M  e.  B )
116 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  n  e.  I )
117105, 113, 114, 115, 116fvmpt2curryd 7018 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  (
(curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) `  n )  =  ( M ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) n ) )
118 df-decpmat 19787 . . . . . . . . . . . . . 14  |- decompPMat  =  ( x  e.  _V , 
k  e.  NN0  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )
119118reseq1i 5101 . . . . . . . . . . . . 13  |-  ( decompPMat  |`  ( B  X.  I ) )  =  ( ( x  e.  _V ,  k  e.  NN0  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )  |`  ( B  X.  I ) )
120 ssv 3452 . . . . . . . . . . . . . . . . 17  |-  B  C_  _V
121120a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  B  C_ 
_V )
122 simpl 459 . . . . . . . . . . . . . . . 16  |-  ( ( I  C_  NN0  /\  I  =/=  (/) )  ->  I  C_ 
NN0 )
123121, 122anim12i 570 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( B  C_  _V  /\  I  C_  NN0 ) )
124123adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  ( B  C_  _V  /\  I  C_ 
NN0 ) )
125 resmpt2 6394 . . . . . . . . . . . . . 14  |-  ( ( B  C_  _V  /\  I  C_ 
NN0 )  ->  (
( x  e.  _V ,  k  e.  NN0  |->  ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )  |`  ( B  X.  I
) )  =  ( x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) )
126124, 125syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  (
( x  e.  _V ,  k  e.  NN0  |->  ( i  e.  dom  dom  x ,  j  e. 
dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )  |`  ( B  X.  I
) )  =  ( x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) )
127119, 126syl5req 2498 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) )  =  ( decompPMat  |`  ( B  X.  I ) ) )
128127oveqd 6307 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  ( M ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) n )  =  ( M ( decompPMat  |`  ( B  X.  I ) ) n ) )
129117, 128eqtrd 2485 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  (
(curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) `  n )  =  ( M ( decompPMat  |`  ( B  X.  I ) ) n ) )
130129adantlr 721 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) ) `  M
) `  n )  =  ( M ( decompPMat  |`  ( B  X.  I
) ) n ) )
131104, 130eqtrd 2485 . . . . . . . 8  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( f `  n
)  =  ( M ( decompPMat  |`  ( B  X.  I ) ) n ) )
132131fveq2d 5869 . . . . . . 7  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( T `  (
f `  n )
)  =  ( T `
 ( M ( decompPMat  |`  ( B  X.  I
) ) n ) ) )
13322ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) )  ->  M  e.  B
)
134 ovres 6436 . . . . . . . . 9  |-  ( ( M  e.  B  /\  n  e.  I )  ->  ( M ( decompPMat  |`  ( B  X.  I ) ) n )  =  ( M decompPMat  n ) )
135133, 134sylan 474 . . . . . . . 8  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( M ( decompPMat  |`  ( B  X.  I ) ) n )  =  ( M decompPMat  n ) )
136135fveq2d 5869 . . . . . . 7  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( T `  ( M ( decompPMat  |`  ( B  X.  I ) ) n ) )  =  ( T `  ( M decompPMat  n ) ) )
137132, 136eqtrd 2485 . . . . . 6  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( T `  (
f `  n )
)  =  ( T `
 ( M decompPMat  n ) ) )
138137oveq2d 6306 . . . . 5  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  (
x  e.  B , 
k  e.  I  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k
) ) ) `  M ) )  /\  n  e.  I )  ->  ( ( n  .^  X )  .*  ( T `  ( f `  n ) ) )  =  ( ( n 
.^  X )  .*  ( T `  ( M decompPMat  n ) ) ) )
139138mpteq2dva 4489 . . . 4  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) )  ->  ( n  e.  I  |->  ( ( n 
.^  X )  .*  ( T `  (
f `  n )
) ) )  =  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( M decompPMat  n ) ) ) ) )
140139oveq2d 6306 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) )  ->  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  (
f `  n )
) ) ) )  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( M decompPMat  n ) ) ) ) ) )
141140eqeq2d 2461 . 2  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  f  =  (curry  ( x  e.  B ,  k  e.  I  |->  ( i  e. 
dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  ( i x j ) ) `  k ) ) ) `
 M ) )  ->  ( M  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( f `  n ) ) ) ) )  <->  M  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( M decompPMat  n ) ) ) ) ) ) )
142101, 141rspcedv 3154 1  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( M  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( M decompPMat  n ) ) ) ) )  ->  E. f  e.  ( D  ^m  I
) M  =  ( C  gsumg  ( n  e.  I  |->  ( ( n  .^  X )  .*  ( T `  ( f `  n ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045   [_csb 3363    C_ wss 3404   (/)c0 3731    |-> cmpt 4461    X. cxp 4832   dom cdm 4834    |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292  curry ccur 7012    ^m cmap 7472   Fincfn 7569   NN0cn0 10869   Basecbs 15121   .scvsca 15194    gsumg cgsu 15339  .gcmg 16672  mulGrpcmgp 17723   CRingccrg 17781  var1cv1 18769  Poly1cpl1 18770  coe1cco1 18771   Mat cmat 19432   matToPolyMat cmat2pmat 19728   decompPMat cdecpmat 19786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-cur 7014  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-hom 15214  df-cco 15215  df-0g 15340  df-prds 15346  df-pws 15348  df-sra 18395  df-rgmod 18396  df-psr 18580  df-opsr 18584  df-psr1 18773  df-ply1 18775  df-coe1 18776  df-dsmm 19295  df-frlm 19310  df-mat 19433  df-decpmat 19787
This theorem is referenced by:  pmatcollpw3  19808  pmatcollpw3fi  19809
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