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Theorem pmatcollpw3lem 19502
Description: Lemma for pmatcollpw3 19503 and pmatcollpw3fi 19504: 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 5213 . . . . . . . . 9  |-  ( x  =  y  ->  dom  x  =  dom  y )
21dmeqd 5215 . . . . . . . 8  |-  ( x  =  y  ->  dom  dom  x  =  dom  dom  y )
3 oveq 6302 . . . . . . . . . 10  |-  ( x  =  y  ->  (
i x j )  =  ( i y j ) )
43fveq2d 5876 . . . . . . . . 9  |-  ( x  =  y  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i y j ) ) )
54fveq1d 5874 . . . . . . . 8  |-  ( x  =  y  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i y j ) ) `  k ) )
62, 2, 5mpt2eq123dv 6358 . . . . . . 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 5872 . . . . . . . 8  |-  ( k  =  l  ->  (
(coe1 `  ( i y j ) ) `  k )  =  ( (coe1 `  ( i y j ) ) `  l ) )
87mpt2eq3dv 6362 . . . . . . 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 6376 . . . . . 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 6730 . . . . . . . . . . 11  |-  ( y  e.  B  ->  dom  y  e.  _V )
11 dmexg 6730 . . . . . . . . . . 11  |-  ( dom  y  e.  _V  ->  dom 
dom  y  e.  _V )
1210, 11syl 16 . . . . . . . . . 10  |-  ( y  e.  B  ->  dom  dom  y  e.  _V )
1312, 12jca 532 . . . . . . . . 9  |-  ( y  e.  B  ->  ( dom  dom  y  e.  _V  /\ 
dom  dom  y  e.  _V ) )
1413ad2antrl 727 . . . . . . . 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 6875 . . . . . . . 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 16 . . . . . . 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 2878 . . . . . 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 757 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  I  =/=  (/) )
19 nn0ex 10822 . . . . . . . 8  |-  NN0  e.  _V
2019ssex 4600 . . . . . . 7  |-  ( I 
C_  NN0  ->  I  e. 
_V )
2120ad2antrl 727 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  I  e.  _V )
22 simp3 998 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  M  e.  B )
2322adantr 465 . . . . . 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 5872 . . . . . . . . 9  |-  ( l  =  k  ->  (
(coe1 `  ( i y j ) ) `  l )  =  ( (coe1 `  ( i y j ) ) `  k ) )
2625mpt2eq3dv 6362 . . . . . . . 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 3843 . . . . . . 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 2466 . . . . . . . . 9  |-  ( x  =  y  <->  y  =  x )
29 eqcom 2466 . . . . . . . . 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 265 . . . . . . . 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 3436 . . . . . . 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 2514 . . . . . 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 4548 . . . . 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 2514 . . . 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 5213 . . . . . . . . . . 11  |-  ( x  =  M  ->  dom  x  =  dom  M )
3635dmeqd 5215 . . . . . . . . . 10  |-  ( x  =  M  ->  dom  dom  x  =  dom  dom  M )
37 oveq 6302 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
i x j )  =  ( i M j ) )
3837fveq2d 5876 . . . . . . . . . . 11  |-  ( x  =  M  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i M j ) ) )
3938fveq1d 5874 . . . . . . . . . 10  |-  ( x  =  M  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
4036, 36, 39mpt2eq123dv 6358 . . . . . . . . 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 466 . . . . . . . 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 3457 . . . . . . 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 2457 . . . . . . . . . . . . 13  |-  ( Base `  P )  =  (
Base `  P )
45 pmatcollpw.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  C
)
4643, 44, 45matbas2i 19142 . . . . . . . . . . . 12  |-  ( M  e.  B  ->  M  e.  ( ( Base `  P
)  ^m  ( N  X.  N ) ) )
47 elmapi 7459 . . . . . . . . . . . 12  |-  ( M  e.  ( ( Base `  P )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  P
) )
48 fdm 5741 . . . . . . . . . . . . . 14  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  dom  M  =  ( N  X.  N ) )
4948dmeqd 5215 . . . . . . . . . . . . 13  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  dom  dom 
M  =  dom  ( N  X.  N ) )
50 dmxpid 5232 . . . . . . . . . . . . 13  |-  dom  ( N  X.  N )  =  N
5149, 50syl6req 2515 . . . . . . . . . . . 12  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  N  =  dom  dom  M )
5246, 47, 513syl 20 . . . . . . . . . . 11  |-  ( M  e.  B  ->  N  =  dom  dom  M )
53523ad2ant3 1019 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  N  =  dom  dom  M )
5453adantr 465 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  N  =  dom  dom  M )
55 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  m  =  M )
5655oveqd 6313 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i m j )  =  ( i M j ) )
5756fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
5857fveq1d 5874 . . . . . . . . 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 6358 . . . . . . . 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 3457 . . . . . . 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 2501 . . . . . 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 465 . . . . 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 4544 . . . 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 2498 . . 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 6302 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
6665adantl 466 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i m j )  =  ( i M j ) )
6766fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
6867fveq1d 5874 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
6968mpt2eq3dv 6362 . . . . . . . 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 3457 . . . . . . 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 725 . . . . . 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 2457 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
74 pmatcollpw3.d . . . . . . 7  |-  D  =  ( Base `  A
)
75 simpll1 1035 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  N  e.  Fin )
76 simpll2 1036 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  R  e.  CRing )
77 simp2 997 . . . . . . . . 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 998 . . . . . . . . 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 465 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  M  e.  B )
80793ad2ant1 1017 . . . . . . . . 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 19146 . . . . . . . 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 3493 . . . . . . . . . . 11  |-  ( I 
C_  NN0  ->  ( k  e.  I  ->  k  e.  NN0 ) )
8382ad2antrl 727 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( k  e.  I  ->  k  e.  NN0 )
)
8483imp 429 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  k  e.  NN0 )
85843ad2ant1 1017 . . . . . . . 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 2457 . . . . . . . . 9  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
87 pmatcollpw.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
8886, 44, 87, 73coe1fvalcl 18469 . . . . . . . 8  |-  ( ( ( i M j )  e.  ( Base `  P )  /\  k  e.  NN0 )  ->  (
(coe1 `  ( i M j ) ) `  k )  e.  (
Base `  R )
)
8981, 85, 88syl2anc 661 . . . . . . 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 19143 . . . . . 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 2545 . . . . 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 2457 . . . . 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 6056 . . . 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 5882 . . . . . . 7  |-  ( Base `  A )  e.  _V
9574, 94eqeltri 2541 . . . . . 6  |-  D  e. 
_V
9695a1i 11 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  D  e.  _V )
9720adantr 465 . . . . 5  |-  ( ( I  C_  NN0  /\  I  =/=  (/) )  ->  I  e.  _V )
98 elmapg 7451 . . . . 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 477 . . . 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 232 . . 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 2545 . 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 5871 . . . . . . . . . . 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 466 . . . . . . . . . 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 465 . . . . . . . . 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 2457 . . . . . . . . . . . 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 6730 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  B  ->  dom  x  e.  _V )
107 dmexg 6730 . . . . . . . . . . . . . . . . 17  |-  ( dom  x  e.  _V  ->  dom 
dom  x  e.  _V )
108106, 107syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  B  ->  dom  dom  x  e.  _V )
109108, 108jca 532 . . . . . . . . . . . . . . 15  |-  ( x  e.  B  ->  ( dom  dom  x  e.  _V  /\ 
dom  dom  x  e.  _V ) )
110109ad2antrl 727 . . . . . . . . . . . . . 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 6875 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 2878 . . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  I  e.  _V )
11523adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  M  e.  B )
116 simpr 461 . . . . . . . . . . . 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 19482 . . . . . . . . . . . . . 14  |- decompPMat  =  ( x  e.  _V , 
k  e.  NN0  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )
119118reseq1i 5279 . . . . . . . . . . . . 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 3519 . . . . . . . . . . . . . . . . 17  |-  B  C_  _V
121120a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  B  C_ 
_V )
122 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( I  C_  NN0  /\  I  =/=  (/) )  ->  I  C_ 
NN0 )
123121, 122anim12i 566 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( B  C_  _V  /\  I  C_  NN0 ) )
124123adantr 465 . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 2511 . . . . . . . . . . . 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 6313 . . . . . . . . . . 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 2498 . . . . . . . . . 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 714 . . . . . . . . 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 2498 . . . . . . . 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 5876 . . . . . . 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 725 . . . . . . . . 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 6441 . . . . . . . . 9  |-  ( ( M  e.  B  /\  n  e.  I )  ->  ( M ( decompPMat  |`  ( B  X.  I ) ) n )  =  ( M decompPMat  n ) )
135133, 134sylan 471 . . . . . . . 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 5876 . . . . . . 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 2498 . . . . . 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 6312 . . . . 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 4543 . . . 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 6312 . . 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 2471 . 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 3214 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109   [_csb 3430    C_ wss 3471   (/)c0 3793    |-> cmpt 4515    X. cxp 5006   dom cdm 5008    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298  curry ccur 7012    ^m cmap 7438   Fincfn 7535   NN0cn0 10816   Basecbs 14735   .scvsca 14807    gsumg cgsu 14949  .gcmg 16274  mulGrpcmgp 17359   CRingccrg 17417  var1cv1 18433  Poly1cpl1 18434  coe1cco1 18435   Mat cmat 19127   matToPolyMat cmat2pmat 19423   decompPMat cdecpmat 19481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-cur 7014  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-ress 14742  df-plusg 14816  df-mulr 14817  df-sca 14819  df-vsca 14820  df-ip 14821  df-tset 14822  df-ple 14823  df-ds 14825  df-hom 14827  df-cco 14828  df-0g 14950  df-prds 14956  df-pws 14958  df-sra 18036  df-rgmod 18037  df-psr 18223  df-opsr 18227  df-psr1 18437  df-ply1 18439  df-coe1 18440  df-dsmm 18981  df-frlm 18996  df-mat 19128  df-decpmat 19482
This theorem is referenced by:  pmatcollpw3  19503  pmatcollpw3fi  19504
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