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Theorem mp2pm2mp 31252
Description: A polynomial over matrices transformed into a matrix consisting of polynomials transformed back into the polynomial 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 )
) ) ) )
mp2pm2mp.m  |-  .x.  =  ( .s `  P )
mp2pm2mp.e  |-  E  =  (.g `  (mulGrp `  P
) )
mp2pm2mp.y  |-  Y  =  (var1 `  R )
mp2pm2mp.i  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
Assertion
Ref Expression
mp2pm2mp  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( T `  ( I `  O ) )  =  O )
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    E, p    L, p    N, p   
i, O, j, k, p    P, p    R, p    Y, p    .x. , p    i, L, j    P, i, j    .x. , k    i, E, j   
i, Y, j    .x. , i,
j    k, E    i, I,
j, k, m    m, L    k, Y    m, O    k, X
Allowed substitution hints:    A( m, p)    B( p)    C( p)    P( m)    Q( i, j, p)    T( i, j, k, m, p)    .x. ( m)    E( m)    .^ ( i, j, p)    F( p)    I( p)    .* ( i, j, p)    X( i, j, p)    Y( m)

Proof of Theorem mp2pm2mp
Dummy variables  b 
l  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmattomply1.a . . . 4  |-  A  =  ( N Mat  R )
2 pmattomply1.q . . . 4  |-  Q  =  (Poly1 `  A )
3 pmattomply1.l . . . 4  |-  L  =  ( Base `  Q
)
4 pmattomply1.p . . . 4  |-  P  =  (Poly1 `  R )
5 mp2pm2mp.m . . . 4  |-  .x.  =  ( .s `  P )
6 mp2pm2mp.e . . . 4  |-  E  =  (.g `  (mulGrp `  P
) )
7 mp2pm2mp.y . . . 4  |-  Y  =  (var1 `  R )
8 mp2pm2mp.i . . . 4  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
9 pmattomply1.c . . . 4  |-  C  =  ( N Mat  P )
10 pmattomply1.b . . . 4  |-  B  =  ( Base `  C
)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10mply1topmatcl 31238 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
I `  O )  e.  B )
12 pmattomply1.f . . . . 5  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
13 oveq 6182 . . . . . . . . . 10  |-  ( m  =  x  ->  (
i m j )  =  ( i x j ) )
1413fveq2d 5779 . . . . . . . . 9  |-  ( m  =  x  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i x j ) ) )
1514fveq1d 5777 . . . . . . . 8  |-  ( m  =  x  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i x j ) ) `  k ) )
1615mpt2eq3dv 6237 . . . . . . 7  |-  ( m  =  x  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i x j ) ) `  k ) ) )
17 fveq2 5775 . . . . . . . 8  |-  ( k  =  y  ->  (
(coe1 `  ( i x j ) ) `  k )  =  ( (coe1 `  ( i x j ) ) `  y ) )
1817mpt2eq3dv 6237 . . . . . . 7  |-  ( k  =  y  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i x j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i x j ) ) `  y ) ) )
1916, 18cbvmpt2v 6251 . . . . . 6  |-  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) )  =  ( x  e.  B ,  y  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i x j ) ) `  y ) ) )
20 oveq 6182 . . . . . . . . . 10  |-  ( x  =  m  ->  (
i x j )  =  ( i m j ) )
2120fveq2d 5779 . . . . . . . . 9  |-  ( x  =  m  ->  (coe1 `  ( i x j ) )  =  (coe1 `  ( i m j ) ) )
2221fveq1d 5777 . . . . . . . 8  |-  ( x  =  m  ->  (
(coe1 `  ( i x j ) ) `  y )  =  ( (coe1 `  ( i m j ) ) `  y ) )
2322mpt2eq3dv 6237 . . . . . . 7  |-  ( x  =  m  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i x j ) ) `  y ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  y ) ) )
24 fveq2 5775 . . . . . . . 8  |-  ( y  =  n  ->  (
(coe1 `  ( i m j ) ) `  y )  =  ( (coe1 `  ( i m j ) ) `  n ) )
2524mpt2eq3dv 6237 . . . . . . 7  |-  ( y  =  n  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  y ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  n ) ) )
2623, 25cbvmpt2v 6251 . . . . . 6  |-  ( x  e.  B ,  y  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i x j ) ) `  y
) ) )  =  ( m  e.  B ,  n  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  n ) ) )
2719, 26eqtri 2478 . . . . 5  |-  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) )  =  ( m  e.  B ,  n  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  n ) ) )
2812, 27eqtri 2478 . . . 4  |-  F  =  ( m  e.  B ,  n  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  n ) ) )
29 pmattomply1.m . . . 4  |-  .*  =  ( .s `  Q )
30 pmattomply1.e . . . 4  |-  .^  =  (.g
`  (mulGrp `  Q )
)
31 pmattomply1.x . . . 4  |-  X  =  (var1 `  A )
32 pmattomply1.t . . . . 5  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )
33 oveq2 6184 . . . . . . . . . 10  |-  ( k  =  n  ->  (
m F k )  =  ( m F n ) )
34 oveq1 6183 . . . . . . . . . 10  |-  ( k  =  n  ->  (
k  .^  X )  =  ( n  .^  X ) )
3533, 34oveq12d 6194 . . . . . . . . 9  |-  ( k  =  n  ->  (
( m F k )  .*  ( k 
.^  X ) )  =  ( ( m F n )  .*  ( n  .^  X
) ) )
3635cbvmptv 4467 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( ( m F k )  .*  ( k  .^  X ) ) )  =  ( n  e. 
NN0  |->  ( ( m F n )  .*  ( n  .^  X
) ) )
3736a1i 11 . . . . . . 7  |-  ( m  e.  B  ->  (
k  e.  NN0  |->  ( ( m F k )  .*  ( k  .^  X ) ) )  =  ( n  e. 
NN0  |->  ( ( m F n )  .*  ( n  .^  X
) ) ) )
3837oveq2d 6192 . . . . . 6  |-  ( m  e.  B  ->  ( Q  gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) )  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( m F n )  .*  (
n  .^  X )
) ) ) )
3938mpteq2ia 4458 . . . . 5  |-  ( m  e.  B  |->  ( Q 
gsumg  ( k  e.  NN0  |->  ( ( m F k )  .*  (
k  .^  X )
) ) ) )  =  ( m  e.  B  |->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( m F n )  .*  ( n  .^  X ) ) ) ) )
4032, 39eqtri 2478 . . . 4  |-  T  =  ( m  e.  B  |->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( m F n )  .*  (
n  .^  X )
) ) ) )
414, 9, 10, 28, 29, 30, 31, 1, 2, 3, 40pmattomply1 31242 . . 3  |-  ( ( I `  O )  e.  B  ->  ( T `  ( I `  O ) )  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) ) )
4211, 41syl 16 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( T `  ( I `  O ) )  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) ) )
431matrng 18426 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
44433adant3 1008 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  A  e.  Ring )
45 eqid 2450 . . . . 5  |-  ( 0g
`  Q )  =  ( 0g `  Q
)
462ply1rng 17796 . . . . . . 7  |-  ( A  e.  Ring  ->  Q  e. 
Ring )
47 rngcmn 16767 . . . . . . 7  |-  ( Q  e.  Ring  ->  Q  e. CMnd
)
4843, 46, 473syl 20 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  Q  e. CMnd )
49483adant3 1008 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  Q  e. CMnd )
50 nn0ex 10672 . . . . . 6  |-  NN0  e.  _V
5150a1i 11 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  NN0  e.  _V )
522ply1lmod 17800 . . . . . . . . . 10  |-  ( A  e.  Ring  ->  Q  e. 
LMod )
5343, 52syl 16 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  Q  e.  LMod )
54533adant3 1008 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  Q  e.  LMod )
5554adantr 465 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  Q  e.  LMod )
56 3simpa 985 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
5756adantr 465 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
5811anim1i 568 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( I `  O
)  e.  B  /\  n  e.  NN0 ) )
59 eqid 2450 . . . . . . . . . 10  |-  ( Base `  A )  =  (
Base `  A )
604, 9, 10, 12, 1, 59pmatcollpw1lem2 31210 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( ( I `  O )  e.  B  /\  n  e.  NN0 ) )  ->  (
( I `  O
) F n )  e.  ( Base `  A
) )
6157, 58, 60syl2anc 661 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( I `  O
) F n )  e.  ( Base `  A
) )
622ply1sca 17801 . . . . . . . . . . . . 13  |-  ( A  e.  Ring  ->  A  =  (Scalar `  Q )
)
6362fveq2d 5779 . . . . . . . . . . . 12  |-  ( A  e.  Ring  ->  ( Base `  A )  =  (
Base `  (Scalar `  Q
) ) )
6463eqcomd 2457 . . . . . . . . . . 11  |-  ( A  e.  Ring  ->  ( Base `  (Scalar `  Q )
)  =  ( Base `  A ) )
6544, 64syl 16 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( Base `  (Scalar `  Q
) )  =  (
Base `  A )
)
6665eleq2d 2519 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
( ( I `  O ) F n )  e.  ( Base `  (Scalar `  Q )
)  <->  ( ( I `
 O ) F n )  e.  (
Base `  A )
) )
6766adantr 465 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( ( I `  O ) F n )  e.  ( Base `  (Scalar `  Q )
)  <->  ( ( I `
 O ) F n )  e.  (
Base `  A )
) )
6861, 67mpbird 232 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( I `  O
) F n )  e.  ( Base `  (Scalar `  Q ) ) )
692, 3, 31, 30, 44mon1ply1 30968 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
n  .^  X )  e.  L )
70 eqid 2450 . . . . . . . 8  |-  (Scalar `  Q )  =  (Scalar `  Q )
71 eqid 2450 . . . . . . . 8  |-  ( Base `  (Scalar `  Q )
)  =  ( Base `  (Scalar `  Q )
)
723, 70, 29, 71lmodvscl 17057 . . . . . . 7  |-  ( ( Q  e.  LMod  /\  (
( I `  O
) F n )  e.  ( Base `  (Scalar `  Q ) )  /\  ( n  .^  X )  e.  L )  -> 
( ( ( I `
 O ) F n )  .*  (
n  .^  X )
)  e.  L )
7355, 68, 69, 72syl3anc 1219 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( ( I `  O ) F n )  .*  ( n 
.^  X ) )  e.  L )
74 eqid 2450 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) )  =  ( n  e. 
NN0  |->  ( ( ( I `  O ) F n )  .*  ( n  .^  X
) ) )
7573, 74fmptd 5952 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) : NN0 --> L )
76 oveq 6182 . . . . . . . . . . 11  |-  ( m  =  b  ->  (
i m j )  =  ( i b j ) )
7776fveq2d 5779 . . . . . . . . . 10  |-  ( m  =  b  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i b j ) ) )
7877fveq1d 5777 . . . . . . . . 9  |-  ( m  =  b  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i b j ) ) `  k ) )
7978mpt2eq3dv 6237 . . . . . . . 8  |-  ( m  =  b  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i b j ) ) `  k ) ) )
80 fveq2 5775 . . . . . . . . 9  |-  ( k  =  n  ->  (
(coe1 `  ( i b j ) ) `  k )  =  ( (coe1 `  ( i b j ) ) `  n ) )
8180mpt2eq3dv 6237 . . . . . . . 8  |-  ( k  =  n  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i b j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i b j ) ) `  n ) ) )
8279, 81cbvmpt2v 6251 . . . . . . 7  |-  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k
) ) )  =  ( b  e.  B ,  n  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i b j ) ) `  n ) ) )
8312, 82eqtri 2478 . . . . . 6  |-  F  =  ( b  e.  B ,  n  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i b j ) ) `  n ) ) )
84 eqid 2450 . . . . . 6  |-  ( b  e.  B  |->  ( Q 
gsumg  ( n  e.  NN0  |->  ( ( b F n )  .*  (
n  .^  X )
) ) ) )  =  ( b  e.  B  |->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( b F n )  .*  ( n  .^  X ) ) ) ) )
85 fveq2 5775 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
(coe1 `  p ) `  k )  =  ( (coe1 `  p ) `  n ) )
8685oveqd 6193 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
i ( (coe1 `  p
) `  k )
j )  =  ( i ( (coe1 `  p
) `  n )
j ) )
87 oveq1 6183 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k E Y )  =  ( n E Y ) )
8886, 87oveq12d 6194 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( i ( (coe1 `  p ) `  k
) j )  .x.  ( k E Y ) )  =  ( ( i ( (coe1 `  p ) `  n
) j )  .x.  ( n E Y ) ) )
8988cbvmptv 4467 . . . . . . . . . . 11  |-  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p
) `  k )
j )  .x.  (
k E Y ) ) )  =  ( n  e.  NN0  |->  ( ( i ( (coe1 `  p
) `  n )
j )  .x.  (
n E Y ) ) )
9089a1i 11 . . . . . . . . . 10  |-  ( ( i  e.  N  /\  j  e.  N )  ->  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) )  =  ( n  e. 
NN0  |->  ( ( i ( (coe1 `  p ) `  n ) j ) 
.x.  ( n E Y ) ) ) )
9190oveq2d 6192 . . . . . . . . 9  |-  ( ( i  e.  N  /\  j  e.  N )  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) )  =  ( P 
gsumg  ( n  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  n ) j ) 
.x.  ( n E Y ) ) ) ) )
9291mpt2eq3ia 6236 . . . . . . . 8  |-  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  =  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( n  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  n ) j ) 
.x.  ( n E Y ) ) ) ) )
9392mpteq2i 4459 . . . . . . 7  |-  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( n  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  n ) j ) 
.x.  ( n E Y ) ) ) ) ) )
948, 93eqtri 2478 . . . . . 6  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( n  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  n ) j ) 
.x.  ( n E Y ) ) ) ) ) )
954, 9, 10, 83, 29, 30, 31, 1, 2, 3, 84, 5, 6, 7, 94mp2pm2mplem5 31251 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) finSupp 
( 0g `  Q
) )
963, 45, 49, 51, 75, 95gsumcl 16487 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) )  e.  L )
97 simp3 990 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  O  e.  L )
9844, 96, 973jca 1168 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( A  e.  Ring  /\  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) )  e.  L  /\  O  e.  L ) )
994, 9, 10, 12, 29, 30, 31, 1, 2, 3, 32, 5, 6, 7, 8mp2pm2mplem4 31250 . . . . . . . . . . 11  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( I `  O
) F n )  =  ( (coe1 `  O
) `  n )
)
10099oveq1d 6191 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  n  e.  NN0 )  ->  (
( ( I `  O ) F n )  .*  ( n 
.^  X ) )  =  ( ( (coe1 `  O ) `  n
)  .*  ( n 
.^  X ) ) )
101100adantlr 714 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  l  e.  NN0 )  /\  n  e. 
NN0 )  ->  (
( ( I `  O ) F n )  .*  ( n 
.^  X ) )  =  ( ( (coe1 `  O ) `  n
)  .*  ( n 
.^  X ) ) )
102101mpteq2dva 4462 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  (
n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) )  =  ( n  e. 
NN0  |->  ( ( (coe1 `  O ) `  n
)  .*  ( n 
.^  X ) ) ) )
103102oveq2d 6192 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) )  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O
) `  n )  .*  ( n  .^  X
) ) ) ) )
104103fveq2d 5779 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) ) )  =  (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O ) `  n )  .*  (
n  .^  X )
) ) ) ) )
105104fveq1d 5777 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  (
(coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) ) ) `  l
)  =  ( (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O
) `  n )  .*  ( n  .^  X
) ) ) ) ) `  l ) )
10644, 97jca 532 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( A  e.  Ring  /\  O  e.  L ) )
107106adantr 465 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  ( A  e.  Ring  /\  O  e.  L ) )
108 eqid 2450 . . . . . . . . . 10  |-  (mulGrp `  Q )  =  (mulGrp `  Q )
109 eqid 2450 . . . . . . . . . 10  |-  (coe1 `  O
)  =  (coe1 `  O
)
1102, 31, 3, 29, 108, 30, 109ply1coe 17841 . . . . . . . . 9  |-  ( ( A  e.  Ring  /\  O  e.  L )  ->  O  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O ) `  n )  .*  (
n  .^  X )
) ) ) )
111107, 110syl 16 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  O  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O ) `  n )  .*  (
n  .^  X )
) ) ) )
112111eqcomd 2457 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O
) `  n )  .*  ( n  .^  X
) ) ) )  =  O )
113112fveq2d 5779 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O
) `  n )  .*  ( n  .^  X
) ) ) ) )  =  (coe1 `  O
) )
114113fveq1d 5777 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  (
(coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  O ) `  n )  .*  (
n  .^  X )
) ) ) ) `
 l )  =  ( (coe1 `  O ) `  l ) )
115105, 114eqtrd 2490 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  l  e.  NN0 )  ->  (
(coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) ) ) `  l
)  =  ( (coe1 `  O ) `  l
) )
116115ralrimiva 2881 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  A. l  e.  NN0  ( (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) ) ) `
 l )  =  ( (coe1 `  O ) `  l ) )
117 eqid 2450 . . . 4  |-  (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) ) )  =  (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) ) )
1182, 3, 117, 109eqcoe1ply1eq 30963 . . 3  |-  ( ( A  e.  Ring  /\  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) )  e.  L  /\  O  e.  L )  ->  ( A. l  e.  NN0  ( (coe1 `  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) ) ) `  l
)  =  ( (coe1 `  O ) `  l
)  ->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `  O
) F n )  .*  ( n  .^  X ) ) ) )  =  O ) )
11998, 116, 118sylc 60 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( I `
 O ) F n )  .*  (
n  .^  X )
) ) )  =  O )
12042, 119eqtrd 2490 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( T `  ( I `  O ) )  =  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   A.wral 2792   _Vcvv 3054    |-> cmpt 4434   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   Fincfn 7396   NN0cn0 10666   Basecbs 14262  Scalarcsca 14329   .scvsca 14330   0gc0g 14466    gsumg cgsu 14467  .gcmg 15502  CMndccmn 16367  mulGrpcmgp 16682   Ringcrg 16737   LModclmod 17040  var1cv1 17725  Poly1cpl1 17726  coe1cco1 17727   Mat cmat 18375
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-ot 3970  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-of 6406  df-ofr 6407  df-om 6563  df-1st 6663  df-2nd 6664  df-supp 6777  df-recs 6918  df-rdg 6952  df-1o 7006  df-2o 7007  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-ixp 7350  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-fsupp 7708  df-sup 7778  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-dec 10843  df-uz 10949  df-fz 11525  df-fzo 11636  df-seq 11894  df-hash 12191  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-sca 14342  df-vsca 14343  df-ip 14344  df-tset 14345  df-ple 14346  df-ds 14348  df-hom 14350  df-cco 14351  df-0g 14468  df-gsum 14469  df-prds 14474  df-pws 14476  df-mre 14612  df-mrc 14613  df-acs 14615  df-mnd 15503  df-mhm 15552  df-submnd 15553  df-grp 15633  df-minusg 15634  df-sbg 15635  df-mulg 15636  df-subg 15766  df-ghm 15833  df-cntz 15923  df-cmn 16369  df-abl 16370  df-mgp 16683  df-ur 16695  df-srg 16699  df-rng 16739  df-subrg 16955  df-lmod 17042  df-lss 17106  df-sra 17345  df-rgmod 17346  df-psr 17515  df-mvr 17516  df-mpl 17517  df-opsr 17519  df-psr1 17729  df-vr1 17730  df-ply1 17731  df-coe1 17732  df-dsmm 18252  df-frlm 18267  df-mamu 18376  df-mat 18377
This theorem is referenced by:  pmattomply1fo  31254
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