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Theorem pmatcollpw3lem 19884
Description: Lemma for pmatcollpw3 19885 and pmatcollpw3fi 19886: 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 5040 . . . . . . . . 9  |-  ( x  =  y  ->  dom  x  =  dom  y )
21dmeqd 5042 . . . . . . . 8  |-  ( x  =  y  ->  dom  dom  x  =  dom  dom  y )
3 oveq 6314 . . . . . . . . . 10  |-  ( x  =  y  ->  (
i x j )  =  ( i y j ) )
43fveq2d 5883 . . . . . . . . 9  |-  ( x  =  y  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i y j ) ) )
54fveq1d 5881 . . . . . . . 8  |-  ( x  =  y  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i y j ) ) `  k ) )
62, 2, 5mpt2eq123dv 6372 . . . . . . 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 5879 . . . . . . . 8  |-  ( k  =  l  ->  (
(coe1 `  ( i y j ) ) `  k )  =  ( (coe1 `  ( i y j ) ) `  l ) )
87mpt2eq3dv 6376 . . . . . . 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 6390 . . . . . 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 6743 . . . . . . . . . . 11  |-  ( y  e.  B  ->  dom  y  e.  _V )
11 dmexg 6743 . . . . . . . . . . 11  |-  ( dom  y  e.  _V  ->  dom 
dom  y  e.  _V )
1210, 11syl 17 . . . . . . . . . 10  |-  ( y  e.  B  ->  dom  dom  y  e.  _V )
1312, 12jca 541 . . . . . . . . 9  |-  ( y  e.  B  ->  ( dom  dom  y  e.  _V  /\ 
dom  dom  y  e.  _V ) )
1413ad2antrl 742 . . . . . . . 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 6888 . . . . . . . 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 2814 . . . . . 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 774 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  I  =/=  (/) )
19 nn0ex 10899 . . . . . . . 8  |-  NN0  e.  _V
2019ssex 4540 . . . . . . 7  |-  ( I 
C_  NN0  ->  I  e. 
_V )
2120ad2antrl 742 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  I  e.  _V )
22 simp3 1032 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  M  e.  B )
2322adantr 472 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  ->  M  e.  B )
249, 17, 18, 21, 23mpt2curryvald 7035 . . . . 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 5879 . . . . . . . . 9  |-  ( l  =  k  ->  (
(coe1 `  ( i y j ) ) `  l )  =  ( (coe1 `  ( i y j ) ) `  k ) )
2625mpt2eq3dv 6376 . . . . . . . 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 3785 . . . . . . 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 2478 . . . . . . . . 9  |-  ( x  =  y  <->  y  =  x )
29 eqcom 2478 . . . . . . . . 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 273 . . . . . . . 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 3355 . . . . . . 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 2521 . . . . . 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 4488 . . . . 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 2521 . . . 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 5040 . . . . . . . . . . 11  |-  ( x  =  M  ->  dom  x  =  dom  M )
3635dmeqd 5042 . . . . . . . . . 10  |-  ( x  =  M  ->  dom  dom  x  =  dom  dom  M )
37 oveq 6314 . . . . . . . . . . . 12  |-  ( x  =  M  ->  (
i x j )  =  ( i M j ) )
3837fveq2d 5883 . . . . . . . . . . 11  |-  ( x  =  M  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i M j ) ) )
3938fveq1d 5881 . . . . . . . . . 10  |-  ( x  =  M  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
4036, 36, 39mpt2eq123dv 6372 . . . . . . . . 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 473 . . . . . . . 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 3376 . . . . . . 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 2471 . . . . . . . . . . . . 13  |-  ( Base `  P )  =  (
Base `  P )
45 pmatcollpw.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  C
)
4643, 44, 45matbas2i 19524 . . . . . . . . . . . 12  |-  ( M  e.  B  ->  M  e.  ( ( Base `  P
)  ^m  ( N  X.  N ) ) )
47 elmapi 7511 . . . . . . . . . . . 12  |-  ( M  e.  ( ( Base `  P )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  P
) )
48 fdm 5745 . . . . . . . . . . . . . 14  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  dom  M  =  ( N  X.  N ) )
4948dmeqd 5042 . . . . . . . . . . . . 13  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  dom  dom 
M  =  dom  ( N  X.  N ) )
50 dmxpid 5060 . . . . . . . . . . . . 13  |-  dom  ( N  X.  N )  =  N
5149, 50syl6req 2522 . . . . . . . . . . . 12  |-  ( M : ( N  X.  N ) --> ( Base `  P )  ->  N  =  dom  dom  M )
5246, 47, 513syl 18 . . . . . . . . . . 11  |-  ( M  e.  B  ->  N  =  dom  dom  M )
53523ad2ant3 1053 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  N  =  dom  dom  M )
5453adantr 472 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  N  =  dom  dom  M )
55 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  m  =  M )
5655oveqd 6325 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i m j )  =  ( i M j ) )
5756fveq2d 5883 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
5857fveq1d 5881 . . . . . . . . 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 6372 . . . . . . . 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 3376 . . . . . . 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 2508 . . . . . 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 472 . . . . 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 4483 . . . 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 2505 . . 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 6314 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
6665adantl 473 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
i m j )  =  ( i M j ) )
6766fveq2d 5883 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
6867fveq1d 5881 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  m  =  M )  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
6968mpt2eq3dv 6376 . . . . . . . 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 3376 . . . . . . 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 740 . . . . . 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 2471 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
74 pmatcollpw3.d . . . . . . 7  |-  D  =  ( Base `  A
)
75 simpll1 1069 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  N  e.  Fin )
76 simpll2 1070 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  R  e.  CRing )
77 simp2 1031 . . . . . . . . 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 1032 . . . . . . . . 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 472 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  M  e.  B )
80793ad2ant1 1051 . . . . . . . . 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 19528 . . . . . . . 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 3412 . . . . . . . . . . 11  |-  ( I 
C_  NN0  ->  ( k  e.  I  ->  k  e.  NN0 ) )
8382ad2antrl 742 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( k  e.  I  ->  k  e.  NN0 )
)
8483imp 436 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  k  e.  I )  ->  k  e.  NN0 )
85843ad2ant1 1051 . . . . . . . 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 2471 . . . . . . . . 9  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
87 pmatcollpw.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
8886, 44, 87, 73coe1fvalcl 18882 . . . . . . . 8  |-  ( ( ( i M j )  e.  ( Base `  P )  /\  k  e.  NN0 )  ->  (
(coe1 `  ( i M j ) ) `  k )  e.  (
Base `  R )
)
8981, 85, 88syl2anc 673 . . . . . . 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 19525 . . . . . 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 2549 . . . . 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 2471 . . . . 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 6061 . . . 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 5889 . . . . . . 7  |-  ( Base `  A )  e.  _V
9574, 94eqeltri 2545 . . . . . 6  |-  D  e. 
_V
9695a1i 11 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  D  e.  _V )
9720adantr 472 . . . . 5  |-  ( ( I  C_  NN0  /\  I  =/=  (/) )  ->  I  e.  _V )
98 elmapg 7503 . . . . 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 485 . . . 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 240 . . 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 2549 . 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 5878 . . . . . . . . . . 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 473 . . . . . . . . . 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 472 . . . . . . . . 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 2471 . . . . . . . . . . . 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 6743 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  B  ->  dom  x  e.  _V )
107 dmexg 6743 . . . . . . . . . . . . . . . . 17  |-  ( dom  x  e.  _V  ->  dom 
dom  x  e.  _V )
108106, 107syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  B  ->  dom  dom  x  e.  _V )
109108, 108jca 541 . . . . . . . . . . . . . . 15  |-  ( x  e.  B  ->  ( dom  dom  x  e.  _V  /\ 
dom  dom  x  e.  _V ) )
110109ad2antrl 742 . . . . . . . . . . . . . 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 6888 . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . 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 472 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  I  e.  _V )
11523adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  ( I  C_ 
NN0  /\  I  =/=  (/) ) )  /\  n  e.  I )  ->  M  e.  B )
116 simpr 468 . . . . . . . . . . . 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 7036 . . . . . . . . . . 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 19864 . . . . . . . . . . . . . 14  |- decompPMat  =  ( x  e.  _V , 
k  e.  NN0  |->  ( i  e.  dom  dom  x ,  j  e.  dom  dom  x  |->  ( (coe1 `  (
i x j ) ) `  k ) ) )
119118reseq1i 5107 . . . . . . . . . . . . 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 3438 . . . . . . . . . . . . . . . . 17  |-  B  C_  _V
121120a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  B  C_ 
_V )
122 simpl 464 . . . . . . . . . . . . . . . 16  |-  ( ( I  C_  NN0  /\  I  =/=  (/) )  ->  I  C_ 
NN0 )
123121, 122anim12i 576 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  C_  NN0  /\  I  =/=  (/) ) )  -> 
( B  C_  _V  /\  I  C_  NN0 ) )
124123adantr 472 . . . . . . . . . . . . . 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 6413 . . . . . . . . . . . . . 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 2518 . . . . . . . . . . . 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 6325 . . . . . . . . . . 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 2505 . . . . . . . . . 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 729 . . . . . . . . 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 2505 . . . . . . . 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 5883 . . . . . . 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 740 . . . . . . . . 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 6455 . . . . . . . . 9  |-  ( ( M  e.  B  /\  n  e.  I )  ->  ( M ( decompPMat  |`  ( B  X.  I ) ) n )  =  ( M decompPMat  n ) )
135133, 134sylan 479 . . . . . . . 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 5883 . . . . . . 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 2505 . . . . . 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 6324 . . . . 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 4482 . . . 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 6324 . . 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 2481 . 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 3140 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031   [_csb 3349    C_ wss 3390   (/)c0 3722    |-> cmpt 4454    X. cxp 4837   dom cdm 4839    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310  curry ccur 7030    ^m cmap 7490   Fincfn 7587   NN0cn0 10893   Basecbs 15199   .scvsca 15272    gsumg cgsu 15417  .gcmg 16750  mulGrpcmgp 17801   CRingccrg 17859  var1cv1 18846  Poly1cpl1 18847  coe1cco1 18848   Mat cmat 19509   matToPolyMat cmat2pmat 19805   decompPMat cdecpmat 19863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-cur 7032  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-0g 15418  df-prds 15424  df-pws 15426  df-sra 18473  df-rgmod 18474  df-psr 18657  df-opsr 18661  df-psr1 18850  df-ply1 18852  df-coe1 18853  df-dsmm 19372  df-frlm 19387  df-mat 19510  df-decpmat 19864
This theorem is referenced by:  pmatcollpw3  19885  pmatcollpw3fi  19886
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