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Theorem decpmatid 19787
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, 2pmatring 19710 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  Ring )
433adant3 1027 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  C  e.  Ring )
5 eqid 2450 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 decpmatid.i . . . . 5  |-  I  =  ( 1r `  C
)
75, 6ringidcl 17794 . . . 4  |-  ( C  e.  Ring  ->  I  e.  ( Base `  C
) )
84, 7syl 17 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  I  e.  ( Base `  C
) )
9 simp3 1009 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  K  e.  NN0 )
102, 5decpmatval 19782 . . 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 666 . 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 2450 . . . . . . 7  |-  ( 0g
`  P )  =  ( 0g `  P
)
13 eqid 2450 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
14 simp11 1037 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  N  e.  Fin )
15 simp12 1038 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
16 simp2 1008 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
17 simp3 1009 . . . . . . 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 19714 . . . . . 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 5867 . . . . 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 5865 . . . 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 5874 . . . . . . 7  |-  (coe1 `  if ( i  =  j ,  ( 1r `  P ) ,  ( 0g `  P ) ) )  =  if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) )
2221fveq1i 5864 . . . . . 6  |-  ( (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) `  K )  =  ( if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) ) `
 K )
23 iffv 5875 . . . . . 6  |-  ( if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) ) `
 K )  =  if ( i  =  j ,  ( (coe1 `  ( 1r `  P
) ) `  K
) ,  ( (coe1 `  ( 0g `  P
) ) `  K
) )
2422, 23eqtri 2472 . . . . 5  |-  ( (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) `  K )  =  if ( i  =  j ,  ( (coe1 `  ( 1r `  P ) ) `
 K ) ,  ( (coe1 `  ( 0g `  P ) ) `  K ) )
25 eqid 2450 . . . . . . . . . . . . 13  |-  (var1 `  R
)  =  (var1 `  R
)
26 eqid 2450 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
27 eqid 2450 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
281, 25, 26, 27ply1idvr1 18879 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( 1r `  P ) )
29283ad2ant2 1029 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( 1r `  P ) )
3029eqcomd 2456 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  P )  =  ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )
3130fveq2d 5867 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 1r `  P ) )  =  (coe1 `  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )
3231fveq1d 5865 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 1r `  P ) ) `  K )  =  ( (coe1 `  ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) `  K ) )
331ply1lmod 18838 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  P  e. 
LMod )
34333ad2ant2 1029 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  P  e.  LMod )
35 0nn0 10881 . . . . . . . . . . . . . 14  |-  0  e.  NN0
36 eqid 2450 . . . . . . . . . . . . . . 15  |-  ( Base `  P )  =  (
Base `  P )
371, 25, 26, 27, 36ply1moncl 18857 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  0  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
3835, 37mpan2 676 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
39383ad2ant2 1029 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
40 eqid 2450 . . . . . . . . . . . . 13  |-  (Scalar `  P )  =  (Scalar `  P )
41 eqid 2450 . . . . . . . . . . . . 13  |-  ( .s
`  P )  =  ( .s `  P
)
42 eqid 2450 . . . . . . . . . . . . 13  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
4336, 40, 41, 42lmodvs1 18112 . . . . . . . . . . . 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 666 . . . . . . . . . . 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 2456 . . . . . . . . . 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 5867 . . . . . . . . 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 5865 . . . . . . . 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 1008 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  R  e.  Ring )
491ply1sca 18839 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  R  =  (Scalar `  P )
)
50493ad2ant2 1029 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  R  =  (Scalar `  P )
)
5150eqcomd 2456 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (Scalar `  P )  =  R )
5251fveq2d 5867 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  (Scalar `  P
) )  =  ( 1r `  R ) )
53 eqid 2450 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
54 eqid 2450 . . . . . . . . . . . . 13  |-  ( 1r
`  R )  =  ( 1r `  R
)
5553, 54ringidcl 17794 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
56553ad2ant2 1029 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  R )  e.  ( Base `  R
) )
5752, 56eqeltrd 2528 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  (Scalar `  P
) )  e.  (
Base `  R )
)
5835a1i 11 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  0  e.  NN0 )
59 eqid 2450 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
6059, 53, 1, 25, 41, 26, 27coe1tm 18859 . . . . . . . . . 10  |-  ( ( 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 1267 . . . . . . . . 9  |-  ( ( 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
) ) ) )
62 eqeq1 2454 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
k  =  0  <->  K  =  0 ) )
6362ifbid 3902 . . . . . . . . . 10  |-  ( k  =  K  ->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  if ( K  =  0 ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )
6463adantl 468 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
65 fvex 5873 . . . . . . . . . . 11  |-  ( 1r
`  (Scalar `  P )
)  e.  _V
66 fvex 5873 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
6765, 66ifex 3948 . . . . . . . . . 10  |-  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) )  e.  _V
6867a1i 11 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  e. 
_V )
6961, 64, 9, 68fvmptd 5952 . . . . . . . 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 ) ) )
7032, 47, 693eqtrd 2488 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 1r `  P ) ) `  K )  =  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) ) )
711, 12, 59coe1z 18849 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  { ( 0g `  R ) } ) )
72713ad2ant2 1029 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0 
X.  { ( 0g
`  R ) } ) )
7372fveq1d 5865 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 0g `  P ) ) `  K )  =  ( ( NN0  X.  {
( 0g `  R
) } ) `  K ) )
7466a1i 11 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 0g `  R )  e. 
_V )
75 fvconst2g 6116 . . . . . . . . 9  |-  ( ( ( 0g `  R
)  e.  _V  /\  K  e.  NN0 )  -> 
( ( NN0  X.  { ( 0g `  R ) } ) `
 K )  =  ( 0g `  R
) )
7674, 9, 75syl2anc 666 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
( NN0  X.  { ( 0g `  R ) } ) `  K
)  =  ( 0g
`  R ) )
7773, 76eqtrd 2484 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 0g `  P ) ) `  K )  =  ( 0g `  R ) )
7870, 77ifeq12d 3900 . . . . . 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 ) ) )
79783ad2ant1 1028 . . . . 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 ) ) )
8024, 79syl5eq 2496 . . . 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 ) ) )
8120, 80eqtrd 2484 . . 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 ) ) )
8281mpt2eq3dva 6352 . 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 ) ) ) )
8350adantl 468 . . . . . . . . 9  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  R  =  (Scalar `  P
) )
8483eqcomd 2456 . . . . . . . 8  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
(Scalar `  P )  =  R )
8584fveq2d 5867 . . . . . . 7  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( 1r `  (Scalar `  P ) )  =  ( 1r `  R
) )
8685ifeq1d 3898 . . . . . 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
) ) )
8786mpt2eq3dv 6354 . . . . 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 ) ) ) )
88 iftrue 3886 . . . . . . . 8  |-  ( K  =  0  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 1r `  (Scalar `  P ) ) )
8988ifeq1d 3898 . . . . . . 7  |-  ( K  =  0  ->  if ( i  =  j ,  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) ,  ( 0g `  R ) )  =  if ( i  =  j ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R
) ) )
9089adantr 467 . . . . . 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
) ) )
9190mpt2eq3dv 6354 . . . . 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 ) ) ) )
92 decpmatid.1 . . . . . . . 8  |-  .1.  =  ( 1r `  A )
93 decpmatid.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
9493, 54, 59mat1 19465 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  A
)  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ) )
9592, 94syl5eq 2496 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
96953adant3 1027 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  .1.  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
9796adantl 468 . . . . 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 ) ) ) )
9887, 91, 973eqtr4d 2494 . . . 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.  )
99 iftrue 3886 . . . . . 6  |-  ( K  =  0  ->  if ( K  =  0 ,  .1.  ,  .0.  )  =  .1.  )
10099eqcomd 2456 . . . . 5  |-  ( K  =  0  ->  .1.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
101100adantr 467 . . . 4  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .1.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
10298, 101eqtrd 2484 . . 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.  ) )
103 ifid 3917 . . . . . . 7  |-  if ( i  =  j ,  ( 0g `  R
) ,  ( 0g
`  R ) )  =  ( 0g `  R )
104103a1i 11 . . . . . 6  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( i  =  j ,  ( 0g `  R ) ,  ( 0g `  R ) )  =  ( 0g
`  R ) )
105104mpt2eq3dv 6354 . . . . 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 ) ) )
106 iffalse 3889 . . . . . . . 8  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
107106adantr 467 . . . . . . 7  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
108107ifeq1d 3898 . . . . . 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 ) ) )
109108mpt2eq3dv 6354 . . . . 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 ) ) ) )
110 3simpa 1004 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
111110adantl 468 . . . . . 6  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
112 decpmatid.0 . . . . . . 7  |-  .0.  =  ( 0g `  A )
11393, 59mat0op 19437 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( i  e.  N ,  j  e.  N  |->  ( 0g
`  R ) ) )
114112, 113syl5eq 2496 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .0.  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R ) ) )
115111, 114syl 17 . . . . 5  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .0.  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R ) ) )
116105, 109, 1153eqtr4d 2494 . . . 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.  )
117 iffalse 3889 . . . . . 6  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  .1.  ,  .0.  )  =  .0.  )
118117eqcomd 2456 . . . . 5  |-  ( -.  K  =  0  ->  .0.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
119118adantr 467 . . . 4  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .0.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
120116, 119eqtrd 2484 . . 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.  ) )
121102, 120pm2.61ian 798 . 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.  ) )
12211, 82, 1213eqtrd 2488 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 371    /\ w3a 984    = wceq 1443    e. wcel 1886   _Vcvv 3044   ifcif 3880   {csn 3967    |-> cmpt 4460    X. cxp 4831   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   Fincfn 7566   0cc0 9536   NN0cn0 10866   Basecbs 15114  Scalarcsca 15186   .scvsca 15187   0gc0g 15331  .gcmg 16665  mulGrpcmgp 17716   1rcur 17728   Ringcrg 17773   LModclmod 18084  var1cv1 18762  Poly1cpl1 18763  coe1cco1 18764   Mat cmat 19425   decompPMat cdecpmat 19779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-ot 3976  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-hom 15207  df-cco 15208  df-0g 15333  df-gsum 15334  df-prds 15339  df-pws 15341  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-ghm 16874  df-cntz 16964  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-subrg 17999  df-lmod 18086  df-lss 18149  df-sra 18388  df-rgmod 18389  df-ascl 18531  df-psr 18573  df-mvr 18574  df-mpl 18575  df-opsr 18577  df-psr1 18766  df-vr1 18767  df-ply1 18768  df-coe1 18769  df-dsmm 19288  df-frlm 19303  df-mamu 19402  df-mat 19426  df-decpmat 19780
This theorem is referenced by:  idpm2idmp  19818
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