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Theorem decpmatid 19078
Description: The matrix consisting of the coefficients in the polynomial entries of the identity matrix is an identity or a zero matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
Hypotheses
Ref Expression
decpmatid.p  |-  P  =  (Poly1 `  R )
decpmatid.c  |-  C  =  ( N Mat  P )
decpmatid.i  |-  I  =  ( 1r `  C
)
decpmatid.a  |-  A  =  ( N Mat  R )
decpmatid.0  |-  .0.  =  ( 0g `  A )
decpmatid.1  |-  .1.  =  ( 1r `  A )
Assertion
Ref Expression
decpmatid  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
I decompPMat  K )  =  if ( K  =  0 ,  .1.  ,  .0.  ) )

Proof of Theorem decpmatid
Dummy variables  i 
j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 decpmatid.p . . . . . 6  |-  P  =  (Poly1 `  R )
2 decpmatid.c . . . . . 6  |-  C  =  ( N Mat  P )
31, 2pmatrng 19001 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  Ring )
433adant3 1016 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  C  e.  Ring )
5 eqid 2467 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 decpmatid.i . . . . 5  |-  I  =  ( 1r `  C
)
75, 6rngidcl 17032 . . . 4  |-  ( C  e.  Ring  ->  I  e.  ( Base `  C
) )
84, 7syl 16 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  I  e.  ( Base `  C
) )
9 simp3 998 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  K  e.  NN0 )
102, 5decpmatval 19073 . . 3  |-  ( ( I  e.  ( Base `  C )  /\  K  e.  NN0 )  ->  (
I decompPMat  K )  =  ( i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i I j ) ) `  K ) ) )
118, 9, 10syl2anc 661 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
I decompPMat  K )  =  ( i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i I j ) ) `  K ) ) )
12 eqid 2467 . . . . . . 7  |-  ( 0g
`  P )  =  ( 0g `  P
)
13 eqid 2467 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
14 simp11 1026 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  N  e.  Fin )
15 simp12 1027 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
16 simp2 997 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
17 simp3 998 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  j  e.  N )
181, 2, 12, 13, 14, 15, 16, 17, 6pmat1ovd 19005 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  (
i I j )  =  if ( i  =  j ,  ( 1r `  P ) ,  ( 0g `  P ) ) )
1918fveq2d 5870 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  (coe1 `  ( i I j ) )  =  (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) )
2019fveq1d 5868 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  (
(coe1 `  ( i I j ) ) `  K )  =  ( (coe1 `  if ( i  =  j ,  ( 1r `  P ) ,  ( 0g `  P ) ) ) `
 K ) )
21 fvif 5877 . . . . . . 7  |-  (coe1 `  if ( i  =  j ,  ( 1r `  P ) ,  ( 0g `  P ) ) )  =  if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) )
2221fveq1i 5867 . . . . . 6  |-  ( (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) `  K )  =  ( if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) ) `
 K )
23 iffv 5878 . . . . . 6  |-  ( if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) ) `
 K )  =  if ( i  =  j ,  ( (coe1 `  ( 1r `  P
) ) `  K
) ,  ( (coe1 `  ( 0g `  P
) ) `  K
) )
2422, 23eqtri 2496 . . . . 5  |-  ( (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) `  K )  =  if ( i  =  j ,  ( (coe1 `  ( 1r `  P ) ) `
 K ) ,  ( (coe1 `  ( 0g `  P ) ) `  K ) )
25 eqid 2467 . . . . . . . . . . . . 13  |-  (var1 `  R
)  =  (var1 `  R
)
26 eqid 2467 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
27 eqid 2467 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
281, 25, 26, 27ply1idvr1 18145 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( 1r `  P ) )
29283ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( 1r `  P ) )
3029eqcomd 2475 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  P )  =  ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )
3130fveq2d 5870 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 1r `  P ) )  =  (coe1 `  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )
3231fveq1d 5868 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 1r `  P ) ) `  K )  =  ( (coe1 `  ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) `  K ) )
331ply1lmod 18104 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  P  e. 
LMod )
34333ad2ant2 1018 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  P  e.  LMod )
35 0nn0 10811 . . . . . . . . . . . . . 14  |-  0  e.  NN0
36 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( Base `  P )  =  (
Base `  P )
371, 25, 26, 27, 36ply1moncl 18123 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  0  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
3835, 37mpan2 671 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
39383ad2ant2 1018 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
40 eqid 2467 . . . . . . . . . . . . 13  |-  (Scalar `  P )  =  (Scalar `  P )
41 eqid 2467 . . . . . . . . . . . . 13  |-  ( .s
`  P )  =  ( .s `  P
)
42 eqid 2467 . . . . . . . . . . . . 13  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
4336, 40, 41, 42lmodvs1 17352 . . . . . . . . . . . 12  |-  ( ( P  e.  LMod  /\  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)  ->  ( ( 1r `  (Scalar `  P
) ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  =  ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )
4434, 39, 43syl2anc 661 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  =  ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) )
4544eqcomd 2475 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( ( 1r `  (Scalar `  P ) ) ( .s `  P ) ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ) )
4645fveq2d 5870 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )  =  (coe1 `  ( ( 1r
`  (Scalar `  P )
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) )
4746fveq1d 5868 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) `  K )  =  ( (coe1 `  (
( 1r `  (Scalar `  P ) ) ( .s `  P ) ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) ) ) `  K ) )
48 simp2 997 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  R  e.  Ring )
491ply1sca 18105 . . . . . . . . . . . . . . 15  |-  ( R  e.  Ring  ->  R  =  (Scalar `  P )
)
50493ad2ant2 1018 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  R  =  (Scalar `  P )
)
5150eqcomd 2475 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (Scalar `  P )  =  R )
5251fveq2d 5870 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  (Scalar `  P
) )  =  ( 1r `  R ) )
53 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  R )  =  (
Base `  R )
54 eqid 2467 . . . . . . . . . . . . . 14  |-  ( 1r
`  R )  =  ( 1r `  R
)
5553, 54rngidcl 17032 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
56553ad2ant2 1018 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  R )  e.  ( Base `  R
) )
5752, 56eqeltrd 2555 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  (Scalar `  P
) )  e.  (
Base `  R )
)
5835a1i 11 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  0  e.  NN0 )
59 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  R )  =  ( 0g `  R
)
6059, 53, 1, 25, 41, 26, 27coe1tm 18125 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  ( 1r `  (Scalar `  P
) )  e.  (
Base `  R )  /\  0  e.  NN0 )  ->  (coe1 `  ( ( 1r
`  (Scalar `  P )
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( k  e. 
NN0  |->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ) )
6148, 57, 58, 60syl3anc 1228 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( ( 1r `  (Scalar `  P ) ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )  =  ( k  e.  NN0  |->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ) )
6261fveq1d 5868 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( ( 1r
`  (Scalar `  P )
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) `
 K )  =  ( ( k  e. 
NN0  |->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ) `  K ) )
63 eqidd 2468 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
k  e.  NN0  |->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) )  =  ( k  e.  NN0  |->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) ) ) )
64 eqeq1 2471 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
k  =  0  <->  K  =  0 ) )
6564ifbid 3961 . . . . . . . . . . 11  |-  ( k  =  K  ->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  if ( K  =  0 ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )
6665adantl 466 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  k  =  K )  ->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  if ( K  =  0 ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )
67 fvex 5876 . . . . . . . . . . . 12  |-  ( 1r
`  (Scalar `  P )
)  e.  _V
68 fvex 5876 . . . . . . . . . . . 12  |-  ( 0g
`  R )  e. 
_V
6967, 68ifex 4008 . . . . . . . . . . 11  |-  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) )  e.  _V
7069a1i 11 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  e. 
_V )
7163, 66, 9, 70fvmptd 5956 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
( k  e.  NN0  |->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) ) ) `
 K )  =  if ( K  =  0 ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )
7262, 71eqtrd 2508 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( ( 1r
`  (Scalar `  P )
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) `
 K )  =  if ( K  =  0 ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )
7332, 47, 723eqtrd 2512 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 1r `  P ) ) `  K )  =  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) ) )
741, 12, 59coe1z 18115 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  { ( 0g `  R ) } ) )
75743ad2ant2 1018 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0 
X.  { ( 0g
`  R ) } ) )
7675fveq1d 5868 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 0g `  P ) ) `  K )  =  ( ( NN0  X.  {
( 0g `  R
) } ) `  K ) )
7768a1i 11 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 0g `  R )  e. 
_V )
78 fvconst2g 6115 . . . . . . . . 9  |-  ( ( ( 0g `  R
)  e.  _V  /\  K  e.  NN0 )  -> 
( ( NN0  X.  { ( 0g `  R ) } ) `
 K )  =  ( 0g `  R
) )
7977, 9, 78syl2anc 661 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
( NN0  X.  { ( 0g `  R ) } ) `  K
)  =  ( 0g
`  R ) )
8076, 79eqtrd 2508 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 0g `  P ) ) `  K )  =  ( 0g `  R ) )
8173, 80ifeq12d 3959 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  if ( i  =  j ,  ( (coe1 `  ( 1r `  P ) ) `
 K ) ,  ( (coe1 `  ( 0g `  P ) ) `  K ) )  =  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )
82813ad2ant1 1017 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  if ( i  =  j ,  ( (coe1 `  ( 1r `  P ) ) `
 K ) ,  ( (coe1 `  ( 0g `  P ) ) `  K ) )  =  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )
8324, 82syl5eq 2520 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  (
(coe1 `  if ( i  =  j ,  ( 1r `  P ) ,  ( 0g `  P ) ) ) `
 K )  =  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )
8420, 83eqtrd 2508 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  (
(coe1 `  ( i I j ) ) `  K )  =  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )
8584mpt2eq3dva 6346 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i I j ) ) `  K ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) ) )
8650adantl 466 . . . . . . . . 9  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  R  =  (Scalar `  P
) )
8786eqcomd 2475 . . . . . . . 8  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
(Scalar `  P )  =  R )
8887fveq2d 5870 . . . . . . 7  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( 1r `  (Scalar `  P ) )  =  ( 1r `  R
) )
8988ifeq1d 3957 . . . . . 6  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( i  =  j ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  if ( i  =  j ,  ( 1r
`  R ) ,  ( 0g `  R
) ) )
9089mpt2eq3dv 6348 . . . . 5  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ) )
91 iftrue 3945 . . . . . . . 8  |-  ( K  =  0  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 1r `  (Scalar `  P ) ) )
9291ifeq1d 3957 . . . . . . 7  |-  ( K  =  0  ->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) )  =  if ( i  =  j ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) )
9392adantr 465 . . . . . 6  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) )  =  if ( i  =  j ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) )
9493mpt2eq3dv 6348 . . . . 5  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) ) ) )
95 decpmatid.1 . . . . . . . 8  |-  .1.  =  ( 1r `  A )
96 decpmatid.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
9796, 54, 59mat1 18756 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  A
)  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ) )
9895, 97syl5eq 2520 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
99983adant3 1016 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  .1.  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
10099adantl 466 . . . . 5  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .1.  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
10190, 94, 1003eqtr4d 2518 . . . 4  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  .1.  )
102 iftrue 3945 . . . . . 6  |-  ( K  =  0  ->  if ( K  =  0 ,  .1.  ,  .0.  )  =  .1.  )
103102eqcomd 2475 . . . . 5  |-  ( K  =  0  ->  .1.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
104103adantr 465 . . . 4  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .1.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
105101, 104eqtrd 2508 . . 3  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
106 ifid 3976 . . . . . . 7  |-  if ( i  =  j ,  ( 0g `  R
) ,  ( 0g
`  R ) )  =  ( 0g `  R )
107106a1i 11 . . . . . 6  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( i  =  j ,  ( 0g `  R ) ,  ( 0g `  R ) )  =  ( 0g
`  R ) )
108107mpt2eq3dv 6348 . . . . 5  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  ( 0g `  R ) ,  ( 0g `  R ) ) )  =  ( i  e.  N , 
j  e.  N  |->  ( 0g `  R ) ) )
109 iffalse 3948 . . . . . . . 8  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
110109adantr 465 . . . . . . 7  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
111110ifeq1d 3957 . . . . . 6  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) )  =  if ( i  =  j ,  ( 0g `  R
) ,  ( 0g
`  R ) ) )
112111mpt2eq3dv 6348 . . . . 5  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  ( 0g `  R ) ,  ( 0g `  R ) ) ) )
113 3simpa 993 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
114113adantl 466 . . . . . 6  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
115 decpmatid.0 . . . . . . 7  |-  .0.  =  ( 0g `  A )
11696, 59mat0op 18728 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( i  e.  N ,  j  e.  N  |->  ( 0g
`  R ) ) )
117115, 116syl5eq 2520 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .0.  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R ) ) )
118114, 117syl 16 . . . . 5  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .0.  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R ) ) )
119108, 112, 1183eqtr4d 2518 . . . 4  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  .0.  )
120 iffalse 3948 . . . . . 6  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  .1.  ,  .0.  )  =  .0.  )
121120eqcomd 2475 . . . . 5  |-  ( -.  K  =  0  ->  .0.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
122121adantr 465 . . . 4  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .0.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
123119, 122eqtrd 2508 . . 3  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
124105, 123pm2.61ian 788 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) ) )  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
12511, 85, 1243eqtrd 2512 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
I decompPMat  K )  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   {csn 4027    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   Fincfn 7517   0cc0 9493   NN0cn0 10796   Basecbs 14493  Scalarcsca 14561   .scvsca 14562   0gc0g 14698  .gcmg 15734  mulGrpcmgp 16955   1rcur 16967   Ringcrg 17012   LModclmod 17324  var1cv1 18026  Poly1cpl1 18027  coe1cco1 18028   Mat cmat 18716   decompPMat cdecpmat 19070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-hom 14582  df-cco 14583  df-0g 14700  df-gsum 14701  df-prds 14706  df-pws 14708  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-mhm 15789  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-mulg 15874  df-subg 16012  df-ghm 16079  df-cntz 16169  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-subrg 17239  df-lmod 17326  df-lss 17391  df-sra 17630  df-rgmod 17631  df-ascl 17774  df-psr 17816  df-mvr 17817  df-mpl 17818  df-opsr 17820  df-psr1 18030  df-vr1 18031  df-ply1 18032  df-coe1 18033  df-dsmm 18570  df-frlm 18585  df-mamu 18693  df-mat 18717  df-decpmat 19071
This theorem is referenced by:  idpm2idmp  19109
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