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Theorem decpmatid 19731
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 19654 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  C  e.  Ring )
433adant3 1025 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  C  e.  Ring )
5 eqid 2420 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 decpmatid.i . . . . 5  |-  I  =  ( 1r `  C
)
75, 6ringidcl 17742 . . . 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 1007 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  K  e.  NN0 )
102, 5decpmatval 19726 . . 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 665 . 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 2420 . . . . . . 7  |-  ( 0g
`  P )  =  ( 0g `  P
)
13 eqid 2420 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
14 simp11 1035 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  N  e.  Fin )
15 simp12 1036 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
16 simp2 1006 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
17 simp3 1007 . . . . . . 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 19658 . . . . . 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 5876 . . . . 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 5874 . . . 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 5883 . . . . . . 7  |-  (coe1 `  if ( i  =  j ,  ( 1r `  P ) ,  ( 0g `  P ) ) )  =  if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) )
2221fveq1i 5873 . . . . . 6  |-  ( (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) `  K )  =  ( if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) ) `
 K )
23 iffv 5884 . . . . . 6  |-  ( if ( i  =  j ,  (coe1 `  ( 1r `  P ) ) ,  (coe1 `  ( 0g `  P ) ) ) `
 K )  =  if ( i  =  j ,  ( (coe1 `  ( 1r `  P
) ) `  K
) ,  ( (coe1 `  ( 0g `  P
) ) `  K
) )
2422, 23eqtri 2449 . . . . 5  |-  ( (coe1 `  if ( i  =  j ,  ( 1r
`  P ) ,  ( 0g `  P
) ) ) `  K )  =  if ( i  =  j ,  ( (coe1 `  ( 1r `  P ) ) `
 K ) ,  ( (coe1 `  ( 0g `  P ) ) `  K ) )
25 eqid 2420 . . . . . . . . . . . . 13  |-  (var1 `  R
)  =  (var1 `  R
)
26 eqid 2420 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
27 eqid 2420 . . . . . . . . . . . . 13  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
281, 25, 26, 27ply1idvr1 18827 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( 1r `  P ) )
29283ad2ant2 1027 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  =  ( 1r `  P ) )
3029eqcomd 2428 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  P )  =  ( 0 (.g `  (mulGrp `  P ) ) (var1 `  R ) ) )
3130fveq2d 5876 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 1r `  P ) )  =  (coe1 `  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )
3231fveq1d 5874 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 1r `  P ) ) `  K )  =  ( (coe1 `  ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) `  K ) )
331ply1lmod 18786 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  P  e. 
LMod )
34333ad2ant2 1027 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  P  e.  LMod )
35 0nn0 10873 . . . . . . . . . . . . . 14  |-  0  e.  NN0
36 eqid 2420 . . . . . . . . . . . . . . 15  |-  ( Base `  P )  =  (
Base `  P )
371, 25, 26, 27, 36ply1moncl 18805 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  0  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
3835, 37mpan2 675 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
39383ad2ant2 1027 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
0 (.g `  (mulGrp `  P
) ) (var1 `  R
) )  e.  (
Base `  P )
)
40 eqid 2420 . . . . . . . . . . . . 13  |-  (Scalar `  P )  =  (Scalar `  P )
41 eqid 2420 . . . . . . . . . . . . 13  |-  ( .s
`  P )  =  ( .s `  P
)
42 eqid 2420 . . . . . . . . . . . . 13  |-  ( 1r
`  (Scalar `  P )
)  =  ( 1r
`  (Scalar `  P )
)
4336, 40, 41, 42lmodvs1 18060 . . . . . . . . . . . 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 665 . . . . . . . . . . 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 2428 . . . . . . . . . 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 5876 . . . . . . . . 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 5874 . . . . . . . 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 1006 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  R  e.  Ring )
491ply1sca 18787 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  R  =  (Scalar `  P )
)
50493ad2ant2 1027 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  R  =  (Scalar `  P )
)
5150eqcomd 2428 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (Scalar `  P )  =  R )
5251fveq2d 5876 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  (Scalar `  P
) )  =  ( 1r `  R ) )
53 eqid 2420 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
54 eqid 2420 . . . . . . . . . . . . 13  |-  ( 1r
`  R )  =  ( 1r `  R
)
5553, 54ringidcl 17742 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
56553ad2ant2 1027 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( 1r `  R )  e.  ( Base `  R
) )
5752, 56eqeltrd 2508 . . . . . . . . . 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 2420 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
6059, 53, 1, 25, 41, 26, 27coe1tm 18807 . . . . . . . . . 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 1264 . . . . . . . . 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 2424 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
k  =  0  <->  K  =  0 ) )
6362ifbid 3928 . . . . . . . . . 10  |-  ( k  =  K  ->  if ( k  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  if ( K  =  0 ,  ( 1r
`  (Scalar `  P )
) ,  ( 0g
`  R ) ) )
6463adantl 467 . . . . . . . . 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 5882 . . . . . . . . . . 11  |-  ( 1r
`  (Scalar `  P )
)  e.  _V
66 fvex 5882 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
6765, 66ifex 3974 . . . . . . . . . 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 5961 . . . . . . . 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 2465 . . . . . . 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 18797 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0  X.  { ( 0g `  R ) } ) )
72713ad2ant2 1027 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (coe1 `  ( 0g `  P ) )  =  ( NN0 
X.  { ( 0g
`  R ) } ) )
7372fveq1d 5874 . . . . . . . 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 6124 . . . . . . . . 9  |-  ( ( ( 0g `  R
)  e.  _V  /\  K  e.  NN0 )  -> 
( ( NN0  X.  { ( 0g `  R ) } ) `
 K )  =  ( 0g `  R
) )
7674, 9, 75syl2anc 665 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
( NN0  X.  { ( 0g `  R ) } ) `  K
)  =  ( 0g
`  R ) )
7773, 76eqtrd 2461 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  (
(coe1 `  ( 0g `  P ) ) `  K )  =  ( 0g `  R ) )
7870, 77ifeq12d 3926 . . . . . 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 1026 . . . . 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 2473 . . . 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 2461 . . 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 6360 . 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 467 . . . . . . . . 9  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  R  =  (Scalar `  P
) )
8483eqcomd 2428 . . . . . . . 8  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
(Scalar `  P )  =  R )
8584fveq2d 5876 . . . . . . 7  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  -> 
( 1r `  (Scalar `  P ) )  =  ( 1r `  R
) )
8685ifeq1d 3924 . . . . . 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 6362 . . . . 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 3912 . . . . . . . 8  |-  ( K  =  0  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 1r `  (Scalar `  P ) ) )
8988ifeq1d 3924 . . . . . . 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 466 . . . . . 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 6362 . . . . 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 19409 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 1r `  A
)  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ) )
9592, 94syl5eq 2473 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  .1.  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
96953adant3 1025 . . . . . 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 467 . . . . 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 2471 . . . 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 3912 . . . . . 6  |-  ( K  =  0  ->  if ( K  =  0 ,  .1.  ,  .0.  )  =  .1.  )
10099eqcomd 2428 . . . . 5  |-  ( K  =  0  ->  .1.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
101100adantr 466 . . . 4  |-  ( ( K  =  0  /\  ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .1.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
10298, 101eqtrd 2461 . . 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 3943 . . . . . . 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 6362 . . . . 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 3915 . . . . . . . 8  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
107106adantr 466 . . . . . . 7  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  if ( K  =  0 ,  ( 1r `  (Scalar `  P ) ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
108107ifeq1d 3924 . . . . . 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 6362 . . . . 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 1002 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  K  e.  NN0 )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
111110adantl 467 . . . . . 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 19381 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( i  e.  N ,  j  e.  N  |->  ( 0g
`  R ) ) )
114112, 113syl5eq 2473 . . . . . 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 2471 . . . 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 3915 . . . . . 6  |-  ( -.  K  =  0  ->  if ( K  =  0 ,  .1.  ,  .0.  )  =  .0.  )
118117eqcomd 2428 . . . . 5  |-  ( -.  K  =  0  ->  .0.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
119118adantr 466 . . . 4  |-  ( ( -.  K  =  0  /\  ( N  e. 
Fin  /\  R  e.  Ring  /\  K  e.  NN0 ) )  ->  .0.  =  if ( K  =  0 ,  .1.  ,  .0.  ) )
120116, 119eqtrd 2461 . . 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 797 . 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 2465 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078   ifcif 3906   {csn 3993    |-> cmpt 4475    X. cxp 4843   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7568   0cc0 9528   NN0cn0 10858   Basecbs 15081  Scalarcsca 15153   .scvsca 15154   0gc0g 15298  .gcmg 16624  mulGrpcmgp 17664   1rcur 17676   Ringcrg 17721   LModclmod 18032  var1cv1 18710  Poly1cpl1 18711  coe1cco1 18712   Mat cmat 19369   decompPMat cdecpmat 19723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-ot 4002  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-fzo 11903  df-seq 12200  df-hash 12502  df-struct 15083  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-plusg 15163  df-mulr 15164  df-sca 15166  df-vsca 15167  df-ip 15168  df-tset 15169  df-ple 15170  df-ds 15172  df-hom 15174  df-cco 15175  df-0g 15300  df-gsum 15301  df-prds 15306  df-pws 15308  df-mre 15444  df-mrc 15445  df-acs 15447  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-mhm 16534  df-submnd 16535  df-grp 16625  df-minusg 16626  df-sbg 16627  df-mulg 16628  df-subg 16766  df-ghm 16833  df-cntz 16923  df-cmn 17373  df-abl 17374  df-mgp 17665  df-ur 17677  df-ring 17723  df-subrg 17947  df-lmod 18034  df-lss 18097  df-sra 18336  df-rgmod 18337  df-ascl 18479  df-psr 18521  df-mvr 18522  df-mpl 18523  df-opsr 18525  df-psr1 18714  df-vr1 18715  df-ply1 18716  df-coe1 18717  df-dsmm 19232  df-frlm 19247  df-mamu 19346  df-mat 19370  df-decpmat 19724
This theorem is referenced by:  idpm2idmp  19762
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