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Theorem pmattomply1fo 31268
Description: The transformation of matrices over polynomials into polynomials over matrices is a function mapping matrices over polynomials onto polynomials over matrices. (Contributed by AV, 12-Oct-2019.)
Hypotheses
Ref Expression
pmattomply1.p  |-  P  =  (Poly1 `  R )
pmattomply1.c  |-  C  =  ( N Mat  P )
pmattomply1.b  |-  B  =  ( Base `  C
)
pmattomply1.f  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
pmattomply1.m  |-  .*  =  ( .s `  Q )
pmattomply1.e  |-  .^  =  (.g
`  (mulGrp `  Q )
)
pmattomply1.x  |-  X  =  (var1 `  A )
pmattomply1.a  |-  A  =  ( N Mat  R )
pmattomply1.q  |-  Q  =  (Poly1 `  A )
pmattomply1.l  |-  L  =  ( Base `  Q
)
pmattomply1.t  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )
Assertion
Ref Expression
pmattomply1fo  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  T : B -onto-> L )
Distinct variable groups:    B, k, m, i, j    i, N, j, k    R, i, j, k    m, F    Q, m    m, X    .* , m    .^ , m    A, i, j, k    C, i, j, k, m   
i, F, j, k   
k, L    m, N    P, k    .* , k    R, m    Q, k    .^ , k    i, L, j, m    P, i, j, m    k, X
Allowed substitution hints:    A( m)    Q( i, j)    T( i, j, k, m)    .^ ( i, j)    .* ( i, j)    X( i, j)

Proof of Theorem pmattomply1fo
Dummy variables  l  u  w  x  y 
z  f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmattomply1.p . . 3  |-  P  =  (Poly1 `  R )
2 pmattomply1.c . . 3  |-  C  =  ( N Mat  P )
3 pmattomply1.b . . 3  |-  B  =  ( Base `  C
)
4 pmattomply1.f . . 3  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
5 pmattomply1.m . . 3  |-  .*  =  ( .s `  Q )
6 pmattomply1.e . . 3  |-  .^  =  (.g
`  (mulGrp `  Q )
)
7 pmattomply1.x . . 3  |-  X  =  (var1 `  A )
8 pmattomply1.a . . 3  |-  A  =  ( N Mat  R )
9 pmattomply1.q . . 3  |-  Q  =  (Poly1 `  A )
10 pmattomply1.l . . 3  |-  L  =  ( Base `  Q
)
11 pmattomply1.t . . 3  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11pmattomply1f 31258 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  T : B --> L )
13 oveq 6196 . . . . . . . . . . 11  |-  ( m  =  w  ->  (
i m j )  =  ( i w j ) )
1413fveq2d 5793 . . . . . . . . . 10  |-  ( m  =  w  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i w j ) ) )
1514fveq1d 5791 . . . . . . . . 9  |-  ( m  =  w  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i w j ) ) `  k ) )
1615mpt2eq3dv 6251 . . . . . . . 8  |-  ( m  =  w  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i w j ) ) `  k ) ) )
17 fveq2 5789 . . . . . . . . 9  |-  ( k  =  u  ->  (
(coe1 `  ( i w j ) ) `  k )  =  ( (coe1 `  ( i w j ) ) `  u ) )
1817mpt2eq3dv 6251 . . . . . . . 8  |-  ( k  =  u  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i w j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i w j ) ) `  u ) ) )
1916, 18cbvmpt2v 6265 . . . . . . 7  |-  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) )  =  ( w  e.  B ,  u  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i w j ) ) `  u ) ) )
20 oveq 6196 . . . . . . . . . . 11  |-  ( w  =  m  ->  (
i w j )  =  ( i m j ) )
2120fveq2d 5793 . . . . . . . . . 10  |-  ( w  =  m  ->  (coe1 `  ( i w j ) )  =  (coe1 `  ( i m j ) ) )
2221fveq1d 5791 . . . . . . . . 9  |-  ( w  =  m  ->  (
(coe1 `  ( i w j ) ) `  u )  =  ( (coe1 `  ( i m j ) ) `  u ) )
2322mpt2eq3dv 6251 . . . . . . . 8  |-  ( w  =  m  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i w j ) ) `  u ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  u ) ) )
24 fveq2 5789 . . . . . . . . . 10  |-  ( u  =  z  ->  (
(coe1 `  ( i m j ) ) `  u )  =  ( (coe1 `  ( i m j ) ) `  z ) )
2524mpt2eq3dv 6251 . . . . . . . . 9  |-  ( u  =  z  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  u ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  z ) ) )
26 oveq1 6197 . . . . . . . . . . . 12  |-  ( i  =  x  ->  (
i m j )  =  ( x m j ) )
2726fveq2d 5793 . . . . . . . . . . 11  |-  ( i  =  x  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( x m j ) ) )
2827fveq1d 5791 . . . . . . . . . 10  |-  ( i  =  x  ->  (
(coe1 `  ( i m j ) ) `  z )  =  ( (coe1 `  ( x m j ) ) `  z ) )
29 oveq2 6198 . . . . . . . . . . . 12  |-  ( j  =  y  ->  (
x m j )  =  ( x m y ) )
3029fveq2d 5793 . . . . . . . . . . 11  |-  ( j  =  y  ->  (coe1 `  ( x m j ) )  =  (coe1 `  ( x m y ) ) )
3130fveq1d 5791 . . . . . . . . . 10  |-  ( j  =  y  ->  (
(coe1 `  ( x m j ) ) `  z )  =  ( (coe1 `  ( x m y ) ) `  z ) )
3228, 31cbvmpt2v 6265 . . . . . . . . 9  |-  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  z
) )  =  ( x  e.  N , 
y  e.  N  |->  ( (coe1 `  ( x m y ) ) `  z ) )
3325, 32syl6eq 2508 . . . . . . . 8  |-  ( u  =  z  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  u ) )  =  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x m y ) ) `  z ) ) )
3423, 33cbvmpt2v 6265 . . . . . . 7  |-  ( w  e.  B ,  u  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  (
i w j ) ) `  u ) ) )  =  ( m  e.  B , 
z  e.  NN0  |->  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x m y ) ) `  z
) ) )
354, 19, 343eqtri 2484 . . . . . 6  |-  F  =  ( m  e.  B ,  z  e.  NN0  |->  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x m y ) ) `  z ) ) )
36 oveq2 6198 . . . . . . . . . . 11  |-  ( k  =  z  ->  (
m F k )  =  ( m F z ) )
37 oveq1 6197 . . . . . . . . . . 11  |-  ( k  =  z  ->  (
k  .^  X )  =  ( z  .^  X ) )
3836, 37oveq12d 6208 . . . . . . . . . 10  |-  ( k  =  z  ->  (
( m F k )  .*  ( k 
.^  X ) )  =  ( ( m F z )  .*  ( z  .^  X
) ) )
3938cbvmptv 4481 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( m F k )  .*  ( k  .^  X ) ) )  =  ( z  e. 
NN0  |->  ( ( m F z )  .*  ( z  .^  X
) ) )
4039oveq2i 6201 . . . . . . . 8  |-  ( Q 
gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) )  =  ( Q  gsumg  ( z  e.  NN0  |->  ( ( m F z )  .*  (
z  .^  X )
) ) )
4140mpteq2i 4473 . . . . . . 7  |-  ( m  e.  B  |->  ( Q 
gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )  =  ( m  e.  B  |->  ( Q  gsumg  ( z  e.  NN0  |->  ( ( m F z )  .*  ( z  .^  X ) ) ) ) )
4211, 41eqtri 2480 . . . . . 6  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( z  e.  NN0  |->  ( ( m F z )  .*  (
z  .^  X )
) ) ) )
43 eqid 2451 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
44 eqid 2451 . . . . . 6  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
45 eqid 2451 . . . . . 6  |-  (var1 `  R
)  =  (var1 `  R
)
46 fveq2 5789 . . . . . . . . . . . . . 14  |-  ( l  =  v  ->  (coe1 `  l )  =  (coe1 `  v ) )
4746fveq1d 5791 . . . . . . . . . . . . 13  |-  ( l  =  v  ->  (
(coe1 `  l ) `  k )  =  ( (coe1 `  v ) `  k ) )
4847oveqd 6207 . . . . . . . . . . . 12  |-  ( l  =  v  ->  (
i ( (coe1 `  l
) `  k )
j )  =  ( i ( (coe1 `  v
) `  k )
j ) )
4948oveq1d 6205 . . . . . . . . . . 11  |-  ( l  =  v  ->  (
( i ( (coe1 `  l ) `  k
) j ) ( .s `  P ) ( k (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  =  ( ( i ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
5049mpteq2dv 4477 . . . . . . . . . 10  |-  ( l  =  v  ->  (
k  e.  NN0  |->  ( ( i ( (coe1 `  l
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( k  e. 
NN0  |->  ( ( i ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )
5150oveq2d 6206 . . . . . . . . 9  |-  ( l  =  v  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )
5251mpt2eq3dv 6251 . . . . . . . 8  |-  ( l  =  v  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  v
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) ) )
53 oveq1 6197 . . . . . . . . . . . 12  |-  ( i  =  x  ->  (
i ( (coe1 `  v
) `  k )
j )  =  ( x ( (coe1 `  v
) `  k )
j ) )
5453oveq1d 6205 . . . . . . . . . . 11  |-  ( i  =  x  ->  (
( i ( (coe1 `  v ) `  k
) j ) ( .s `  P ) ( k (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  =  ( ( x ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
5554mpteq2dv 4477 . . . . . . . . . 10  |-  ( i  =  x  ->  (
k  e.  NN0  |->  ( ( i ( (coe1 `  v
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( k  e. 
NN0  |->  ( ( x ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )
5655oveq2d 6206 . . . . . . . . 9  |-  ( i  =  x  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )  =  ( P  gsumg  ( k  e.  NN0  |->  ( ( x ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )
57 oveq2 6198 . . . . . . . . . . . . 13  |-  ( j  =  y  ->  (
x ( (coe1 `  v
) `  k )
j )  =  ( x ( (coe1 `  v
) `  k )
y ) )
5857oveq1d 6205 . . . . . . . . . . . 12  |-  ( j  =  y  ->  (
( x ( (coe1 `  v ) `  k
) j ) ( .s `  P ) ( k (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  =  ( ( x ( (coe1 `  v ) `  k ) y ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
5958mpteq2dv 4477 . . . . . . . . . . 11  |-  ( j  =  y  ->  (
k  e.  NN0  |->  ( ( x ( (coe1 `  v
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( k  e. 
NN0  |->  ( ( x ( (coe1 `  v ) `  k ) y ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )
60 fveq2 5789 . . . . . . . . . . . . . 14  |-  ( k  =  z  ->  (
(coe1 `  v ) `  k )  =  ( (coe1 `  v ) `  z ) )
6160oveqd 6207 . . . . . . . . . . . . 13  |-  ( k  =  z  ->  (
x ( (coe1 `  v
) `  k )
y )  =  ( x ( (coe1 `  v
) `  z )
y ) )
62 oveq1 6197 . . . . . . . . . . . . 13  |-  ( k  =  z  ->  (
k (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( z (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )
6361, 62oveq12d 6208 . . . . . . . . . . . 12  |-  ( k  =  z  ->  (
( x ( (coe1 `  v ) `  k
) y ) ( .s `  P ) ( k (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  =  ( ( x ( (coe1 `  v ) `  z ) y ) ( .s `  P
) ( z (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
6463cbvmptv 4481 . . . . . . . . . . 11  |-  ( k  e.  NN0  |->  ( ( x ( (coe1 `  v
) `  k )
y ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( z  e. 
NN0  |->  ( ( x ( (coe1 `  v ) `  z ) y ) ( .s `  P
) ( z (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
6559, 64syl6eq 2508 . . . . . . . . . 10  |-  ( j  =  y  ->  (
k  e.  NN0  |->  ( ( x ( (coe1 `  v
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( z  e. 
NN0  |->  ( ( x ( (coe1 `  v ) `  z ) y ) ( .s `  P
) ( z (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )
6665oveq2d 6206 . . . . . . . . 9  |-  ( j  =  y  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( x ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )  =  ( P  gsumg  ( z  e.  NN0  |->  ( ( x ( (coe1 `  v ) `  z ) y ) ( .s `  P
) ( z (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )
6756, 66cbvmpt2v 6265 . . . . . . . 8  |-  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  v ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )  =  ( x  e.  N ,  y  e.  N  |->  ( P  gsumg  ( z  e.  NN0  |->  ( ( x ( (coe1 `  v
) `  z )
y ) ( .s
`  P ) ( z (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) )
6852, 67syl6eq 2508 . . . . . . 7  |-  ( l  =  v  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )  =  ( x  e.  N ,  y  e.  N  |->  ( P  gsumg  ( z  e.  NN0  |->  ( ( x ( (coe1 `  v
) `  z )
y ) ( .s
`  P ) ( z (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) ) )
6968cbvmptv 4481 . . . . . 6  |-  ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) )  =  ( v  e.  L  |->  ( x  e.  N ,  y  e.  N  |->  ( P 
gsumg  ( z  e.  NN0  |->  ( ( x ( (coe1 `  v ) `  z ) y ) ( .s `  P
) ( z (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) )
701, 2, 3, 35, 5, 6, 7, 8, 9, 10, 42, 43, 44, 45, 69mp2pm2mp 31266 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  p  e.  L )  ->  ( T `  ( (
l  e.  L  |->  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) )  =  p )
71703expa 1188 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  p  e.  L
)  ->  ( T `  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) ) ) `  p ) )  =  p )
72 eqid 2451 . . . . . . . . 9  |-  ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) )  =  ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) )
738, 9, 10, 1, 43, 44, 45, 72, 2, 3mply1topmatcl 31252 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  p  e.  L )  ->  (
( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p )  e.  B )
74733expa 1188 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  p  e.  L
)  ->  ( (
l  e.  L  |->  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p )  e.  B )
75 simpr 461 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  p  e.  L )  /\  f  =  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) )  ->  f  =  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) ) ) `  p ) )
7675fveq2d 5793 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  p  e.  L )  /\  f  =  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) )  ->  ( T `  f )  =  ( T `  ( ( l  e.  L  |->  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) ) )
7776eqeq2d 2465 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  p  e.  L )  /\  f  =  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) )  ->  ( p  =  ( T `  f )  <->  p  =  ( T `  ( ( l  e.  L  |->  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) ) ) )
7874, 77rspcedv 3173 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  p  e.  L
)  ->  ( p  =  ( T `  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) ) ) `  p ) )  ->  E. f  e.  B  p  =  ( T `  f ) ) )
7978com12 31 . . . . 5  |-  ( p  =  ( T `  ( ( l  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l
) `  k )
j ) ( .s
`  P ) ( k (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) ) ) `  p ) )  -> 
( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  p  e.  L )  ->  E. f  e.  B  p  =  ( T `  f ) ) )
8079eqcoms 2463 . . . 4  |-  ( ( T `  ( ( l  e.  L  |->  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  l ) `  k ) j ) ( .s `  P
) ( k (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) ) `  p ) )  =  p  -> 
( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  p  e.  L )  ->  E. f  e.  B  p  =  ( T `  f ) ) )
8171, 80mpcom 36 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  p  e.  L
)  ->  E. f  e.  B  p  =  ( T `  f ) )
8281ralrimiva 2822 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A. p  e.  L  E. f  e.  B  p  =  ( T `  f ) )
83 dffo3 5957 . 2  |-  ( T : B -onto-> L  <->  ( T : B --> L  /\  A. p  e.  L  E. f  e.  B  p  =  ( T `  f ) ) )
8412, 82, 83sylanbrc 664 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  T : B -onto-> L )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    |-> cmpt 4448   -->wf 5512   -onto->wfo 5514   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   Fincfn 7410   NN0cn0 10680   Basecbs 14276   .scvsca 14344    gsumg cgsu 14481  .gcmg 15516  mulGrpcmgp 16696   Ringcrg 16751  var1cv1 17739  Poly1cpl1 17740  coe1cco1 17741   Mat cmat 18389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-ofr 6421  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-pws 14490  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-srg 16713  df-rng 16753  df-subrg 16969  df-lmod 17056  df-lss 17120  df-sra 17359  df-rgmod 17360  df-psr 17529  df-mvr 17530  df-mpl 17531  df-opsr 17533  df-psr1 17743  df-vr1 17744  df-ply1 17745  df-coe1 17746  df-dsmm 18266  df-frlm 18281  df-mamu 18390  df-mat 18391
This theorem is referenced by:  pmattomply1f1o  31269
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