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Theorem pmatcollpwscmatlem2 19813
Description: Lemma 2 for pmatcollpwscmat 19814. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpwscmat.p  |-  P  =  (Poly1 `  R )
pmatcollpwscmat.c  |-  C  =  ( N Mat  P )
pmatcollpwscmat.b  |-  B  =  ( Base `  C
)
pmatcollpwscmat.m1  |-  .*  =  ( .s `  C )
pmatcollpwscmat.e1  |-  .^  =  (.g
`  (mulGrp `  P )
)
pmatcollpwscmat.x  |-  X  =  (var1 `  R )
pmatcollpwscmat.t  |-  T  =  ( N matToPolyMat  R )
pmatcollpwscmat.a  |-  A  =  ( N Mat  R )
pmatcollpwscmat.d  |-  D  =  ( Base `  A
)
pmatcollpwscmat.u  |-  U  =  (algSc `  P )
pmatcollpwscmat.k  |-  K  =  ( Base `  R
)
pmatcollpwscmat.e2  |-  E  =  ( Base `  P
)
pmatcollpwscmat.s  |-  S  =  (algSc `  P )
pmatcollpwscmat.1  |-  .1.  =  ( 1r `  C )
pmatcollpwscmat.m2  |-  M  =  ( Q  .*  .1.  )
Assertion
Ref Expression
pmatcollpwscmatlem2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( T `  ( M decompPMat  L )
)  =  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  ) )

Proof of Theorem pmatcollpwscmatlem2
Dummy variables  a 
b  i  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 458 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
2 simpr 462 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  e.  Ring )
32adantr 466 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  R  e.  Ring )
4 simpr 462 . . . . . . . 8  |-  ( ( L  e.  NN0  /\  Q  e.  E )  ->  Q  e.  E )
54anim2i 571 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  Q  e.  E ) )
6 df-3an 984 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  <->  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  Q  e.  E ) )
75, 6sylibr 215 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( N  e.  Fin  /\  R  e. 
Ring  /\  Q  e.  E
) )
8 pmatcollpwscmat.m2 . . . . . . 7  |-  M  =  ( Q  .*  .1.  )
9 pmatcollpwscmat.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
10 pmatcollpwscmat.c . . . . . . . 8  |-  C  =  ( N Mat  P )
11 pmatcollpwscmat.b . . . . . . . 8  |-  B  =  ( Base `  C
)
12 pmatcollpwscmat.e2 . . . . . . . 8  |-  E  =  ( Base `  P
)
13 pmatcollpwscmat.m1 . . . . . . . 8  |-  .*  =  ( .s `  C )
14 pmatcollpwscmat.1 . . . . . . . 8  |-  .1.  =  ( 1r `  C )
159, 10, 11, 12, 13, 141pmatscmul 19725 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  ->  ( Q  .*  .1.  )  e.  B )
168, 15syl5eqel 2511 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  ->  M  e.  B )
177, 16syl 17 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  M  e.  B )
18 simprl 762 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  L  e.  NN0 )
19 pmatcollpwscmat.a . . . . . 6  |-  A  =  ( N Mat  R )
20 pmatcollpwscmat.d . . . . . 6  |-  D  =  ( Base `  A
)
219, 10, 11, 19, 20decpmatcl 19790 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B  /\  L  e. 
NN0 )  ->  ( M decompPMat  L )  e.  D
)
223, 17, 18, 21syl3anc 1264 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( M decompPMat  L )  e.  D )
23 df-3an 984 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( M decompPMat  L )  e.  D
)  <->  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M decompPMat  L )  e.  D
) )
241, 22, 23sylanbrc 668 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( N  e.  Fin  /\  R  e. 
Ring  /\  ( M decompPMat  L )  e.  D ) )
25 pmatcollpwscmat.t . . . 4  |-  T  =  ( N matToPolyMat  R )
26 eqid 2422 . . . 4  |-  (algSc `  P )  =  (algSc `  P )
2725, 19, 20, 9, 26mat2pmatval 19747 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( M decompPMat  L )  e.  D
)  ->  ( T `  ( M decompPMat  L )
)  =  ( i  e.  N ,  j  e.  N  |->  ( (algSc `  P ) `  (
i ( M decompPMat  L ) j ) ) ) )
2824, 27syl 17 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( T `  ( M decompPMat  L )
)  =  ( i  e.  N ,  j  e.  N  |->  ( (algSc `  P ) `  (
i ( M decompPMat  L ) j ) ) ) )
293, 17, 183jca 1185 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( R  e.  Ring  /\  M  e.  B  /\  L  e.  NN0 ) )
30293ad2ant1 1026 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( R  e.  Ring  /\  M  e.  B  /\  L  e.  NN0 ) )
31 3simpc 1004 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( i  e.  N  /\  j  e.  N
) )
329, 10, 11decpmate 19789 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B  /\  L  e.  NN0 )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  (
i ( M decompPMat  L ) j )  =  ( (coe1 `  ( i M j ) ) `  L ) )
3330, 31, 32syl2anc 665 . . . 4  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( i ( M decompPMat  L ) j )  =  ( (coe1 `  (
i M j ) ) `  L ) )
3433fveq2d 5886 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( (algSc `  P
) `  ( i
( M decompPMat  L ) j ) )  =  ( (algSc `  P ) `  ( (coe1 `  ( i M j ) ) `  L ) ) )
3534mpt2eq3dva 6370 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( i  e.  N ,  j  e.  N  |->  ( (algSc `  P ) `  (
i ( M decompPMat  L ) j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (algSc `  P ) `  (
(coe1 `  ( i M j ) ) `  L ) ) ) )
36 simp1lr 1069 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
37 simp2 1006 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
38 simp3 1007 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  j  e.  N )
39173ad2ant1 1026 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  M  e.  B )
4010, 12, 11, 37, 38, 39matecld 19450 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( i M j )  e.  E )
41183ad2ant1 1026 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  L  e.  NN0 )
42 eqid 2422 . . . . . . 7  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
43 pmatcollpwscmat.k . . . . . . 7  |-  K  =  ( Base `  R
)
4442, 12, 9, 43coe1fvalcl 18805 . . . . . 6  |-  ( ( ( i M j )  e.  E  /\  L  e.  NN0 )  -> 
( (coe1 `  ( i M j ) ) `  L )  e.  K
)
4540, 41, 44syl2anc 665 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( (coe1 `  ( i M j ) ) `  L )  e.  K
)
46 eqid 2422 . . . . . 6  |-  (var1 `  R
)  =  (var1 `  R
)
47 eqid 2422 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
48 eqid 2422 . . . . . 6  |-  (mulGrp `  P )  =  (mulGrp `  P )
49 eqid 2422 . . . . . 6  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
5043, 9, 46, 47, 48, 49, 26ply1scltm 18874 . . . . 5  |-  ( ( R  e.  Ring  /\  (
(coe1 `  ( i M j ) ) `  L )  e.  K
)  ->  ( (algSc `  P ) `  (
(coe1 `  ( i M j ) ) `  L ) )  =  ( ( (coe1 `  (
i M j ) ) `  L ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
5136, 45, 50syl2anc 665 . . . 4  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( (algSc `  P
) `  ( (coe1 `  ( i M j ) ) `  L
) )  =  ( ( (coe1 `  ( i M j ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )
5251mpt2eq3dva 6370 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( i  e.  N ,  j  e.  N  |->  ( (algSc `  P ) `  (
(coe1 `  ( i M j ) ) `  L ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) )
53 pmatcollpwscmat.e1 . . . . . . 7  |-  .^  =  (.g
`  (mulGrp `  P )
)
54 pmatcollpwscmat.x . . . . . . 7  |-  X  =  (var1 `  R )
55 pmatcollpwscmat.u . . . . . . 7  |-  U  =  (algSc `  P )
56 pmatcollpwscmat.s . . . . . . 7  |-  S  =  (algSc `  P )
579, 10, 11, 13, 53, 54, 25, 19, 20, 55, 43, 12, 56, 14, 8pmatcollpwscmatlem1 19812 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  (
( (coe1 `  ( a M b ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  =  if ( a  =  b ,  ( U `
 ( (coe1 `  Q
) `  L )
) ,  ( 0g
`  P ) ) )
58 eqidd 2423 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  (
i  e.  N , 
j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) )
59 oveq12 6315 . . . . . . . . . . 11  |-  ( ( i  =  a  /\  j  =  b )  ->  ( i M j )  =  ( a M b ) )
6059fveq2d 5886 . . . . . . . . . 10  |-  ( ( i  =  a  /\  j  =  b )  ->  (coe1 `  ( i M j ) )  =  (coe1 `  ( a M b ) ) )
6160fveq1d 5884 . . . . . . . . 9  |-  ( ( i  =  a  /\  j  =  b )  ->  ( (coe1 `  ( i M j ) ) `  L )  =  ( (coe1 `  ( a M b ) ) `  L ) )
6261oveq1d 6321 . . . . . . . 8  |-  ( ( i  =  a  /\  j  =  b )  ->  ( ( (coe1 `  (
i M j ) ) `  L ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) )  =  ( ( (coe1 `  ( a M b ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )
6362adantl 467 . . . . . . 7  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  /\  (
i  =  a  /\  j  =  b )
)  ->  ( (
(coe1 `  ( i M j ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  =  ( ( (coe1 `  (
a M b ) ) `  L ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
64 simprl 762 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  a  e.  N )
65 simprr 764 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  b  e.  N )
66 ovex 6334 . . . . . . . 8  |-  ( ( (coe1 `  ( a M b ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  e. 
_V
6766a1i 11 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  (
( (coe1 `  ( a M b ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  e. 
_V )
6858, 63, 64, 65, 67ovmpt2d 6439 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  (
a ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) b )  =  ( ( (coe1 `  ( a M b ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )
69 simpll 758 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  N  e.  Fin )
709ply1ring 18841 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
7170adantl 467 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  P  e.  Ring )
7271adantr 466 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  P  e.  Ring )
73 pm3.22 450 . . . . . . . . . . 11  |-  ( ( L  e.  NN0  /\  Q  e.  E )  ->  ( Q  e.  E  /\  L  e.  NN0 ) )
7473adantl 467 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( Q  e.  E  /\  L  e. 
NN0 ) )
75 eqid 2422 . . . . . . . . . . 11  |-  (coe1 `  Q
)  =  (coe1 `  Q
)
7675, 12, 9, 43coe1fvalcl 18805 . . . . . . . . . 10  |-  ( ( Q  e.  E  /\  L  e.  NN0 )  -> 
( (coe1 `  Q ) `  L )  e.  K
)
7774, 76syl 17 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( (coe1 `  Q ) `  L
)  e.  K )
789, 55, 43, 12ply1sclcl 18879 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
(coe1 `  Q ) `  L )  e.  K
)  ->  ( U `  ( (coe1 `  Q ) `  L ) )  e.  E )
793, 77, 78syl2anc 665 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( U `  ( (coe1 `  Q ) `  L ) )  e.  E )
8069, 72, 793jca 1185 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( N  e.  Fin  /\  P  e. 
Ring  /\  ( U `  ( (coe1 `  Q ) `  L ) )  e.  E ) )
81 eqid 2422 . . . . . . . 8  |-  ( 0g
`  P )  =  ( 0g `  P
)
8210, 12, 81, 14, 13scmatscmide 19531 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  P  e.  Ring  /\  ( U `  ( (coe1 `  Q ) `  L
) )  e.  E
)  /\  ( a  e.  N  /\  b  e.  N ) )  -> 
( a ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  ) b )  =  if ( a  =  b ,  ( U `
 ( (coe1 `  Q
) `  L )
) ,  ( 0g
`  P ) ) )
8380, 82sylan 473 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  (
a ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )
b )  =  if ( a  =  b ,  ( U `  ( (coe1 `  Q ) `  L ) ) ,  ( 0g `  P
) ) )
8457, 68, 833eqtr4d 2473 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  (
a ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) b )  =  ( a ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )
b ) )
8584ralrimivva 2843 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  A. a  e.  N  A. b  e.  N  ( a
( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  (
i M j ) ) `  L ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) b )  =  ( a ( ( U `  (
(coe1 `  Q ) `  L ) )  .*  .1.  ) b ) )
86 0nn0 10892 . . . . . . . 8  |-  0  e.  NN0
8786a1i 11 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  0  e.  NN0 )
8843, 9, 46, 47, 48, 49, 12ply1tmcl 18865 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
(coe1 `  ( i M j ) ) `  L )  e.  K  /\  0  e.  NN0 )  ->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  e.  E )
8936, 45, 87, 88syl3anc 1264 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( ( (coe1 `  (
i M j ) ) `  L ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) )  e.  E )
9010, 12, 11, 69, 72, 89matbas2d 19447 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  e.  B )
919, 10, 11, 12, 13, 141pmatscmul 19725 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( U `  ( (coe1 `  Q ) `  L
) )  e.  E
)  ->  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  )  e.  B )
9269, 3, 79, 91syl3anc 1264 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  )  e.  B )
9310, 11eqmat 19448 . . . . 5  |-  ( ( ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  (
i M j ) ) `  L ) ( .s `  P
) ( 0 (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )  e.  B  /\  ( ( U `  ( (coe1 `  Q ) `  L ) )  .*  .1.  )  e.  B
)  ->  ( (
i  e.  N , 
j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )  <->  A. a  e.  N  A. b  e.  N  (
a ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) b )  =  ( a ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )
b ) ) )
9490, 92, 93syl2anc 665 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( (
i  e.  N , 
j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L ) ( .s
`  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )  <->  A. a  e.  N  A. b  e.  N  (
a ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) b )  =  ( a ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )
b ) ) )
9585, 94mpbird 235 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( i  e.  N ,  j  e.  N  |->  ( ( (coe1 `  ( i M j ) ) `  L
) ( .s `  P ) ( 0 (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) )  =  ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )
)
9652, 95eqtrd 2463 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( i  e.  N ,  j  e.  N  |->  ( (algSc `  P ) `  (
(coe1 `  ( i M j ) ) `  L ) ) )  =  ( ( U `
 ( (coe1 `  Q
) `  L )
)  .*  .1.  )
)
9728, 35, 963eqtrd 2467 1  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( T `  ( M decompPMat  L )
)  =  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   _Vcvv 3080   ifcif 3911   ` cfv 5601  (class class class)co 6306    |-> cmpt2 6308   Fincfn 7581   0cc0 9547   NN0cn0 10877   Basecbs 15121   .scvsca 15194   0gc0g 15338  .gcmg 16672  mulGrpcmgp 17723   1rcur 17735   Ringcrg 17780  algSccascl 18535  var1cv1 18769  Poly1cpl1 18770  coe1cco1 18771   Mat cmat 19431   matToPolyMat cmat2pmat 19727   decompPMat cdecpmat 19785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-ot 4007  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-ofr 6547  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-ixp 7535  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-sup 7966  df-oi 8035  df-card 8382  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-fz 11793  df-fzo 11924  df-seq 12221  df-hash 12523  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-hom 15214  df-cco 15215  df-0g 15340  df-gsum 15341  df-prds 15346  df-pws 15348  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-subrg 18006  df-lmod 18093  df-lss 18156  df-sra 18395  df-rgmod 18396  df-ascl 18538  df-psr 18580  df-mvr 18581  df-mpl 18582  df-opsr 18584  df-psr1 18773  df-vr1 18774  df-ply1 18775  df-coe1 18776  df-dsmm 19294  df-frlm 19309  df-mamu 19408  df-mat 19432  df-mat2pmat 19730  df-decpmat 19786
This theorem is referenced by:  pmatcollpwscmat  19814
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