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Theorem pmatcollpwscmatlem2 19053
Description: Lemma 2 for pmatcollpwscmat 19054. (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 457 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
2 simpr 461 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  R  e.  Ring )
32adantr 465 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  R  e.  Ring )
4 simpr 461 . . . . . . . 8  |-  ( ( L  e.  NN0  /\  Q  e.  E )  ->  Q  e.  E )
54anim2i 569 . . . . . . 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 970 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  <->  ( ( N  e.  Fin  /\  R  e.  Ring )  /\  Q  e.  E ) )
75, 6sylibr 212 . . . . . 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 18965 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  ->  ( Q  .*  .1.  )  e.  B )
168, 15syl5eqel 2554 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  ->  M  e.  B )
177, 16syl 16 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  M  e.  B )
18 simprl 755 . . . . 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 19030 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B  /\  L  e. 
NN0 )  ->  ( M decompPMat  L )  e.  D
)
223, 17, 18, 21syl3anc 1223 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( M decompPMat  L )  e.  D )
23 df-3an 970 . . . 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 664 . . 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 2462 . . . 4  |-  (algSc `  P )  =  (algSc `  P )
2725, 19, 20, 9, 26mat2pmatval 18987 . . 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 16 . 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 1171 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( R  e.  Ring  /\  M  e.  B  /\  L  e.  NN0 ) )
30293ad2ant1 1012 . . . . 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 990 . . . . 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 19029 . . . . 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 661 . . . 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 5863 . . 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 6338 . 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 1055 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
37 simp2 992 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
38 simp3 993 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  j  e.  N )
39173ad2ant1 1012 . . . . . . 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 18690 . . . . . 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 1012 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  L  e.  NN0 )
42 eqid 2462 . . . . . . 7  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
43 pmatcollpwscmat.k . . . . . . 7  |-  K  =  ( Base `  R
)
4442, 12, 9, 43coe1fvalcl 18017 . . . . . 6  |-  ( ( ( i M j )  e.  E  /\  L  e.  NN0 )  -> 
( (coe1 `  ( i M j ) ) `  L )  e.  K
)
4540, 41, 44syl2anc 661 . . . . 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 2462 . . . . . 6  |-  (var1 `  R
)  =  (var1 `  R
)
47 eqid 2462 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
48 eqid 2462 . . . . . 6  |-  (mulGrp `  P )  =  (mulGrp `  P )
49 eqid 2462 . . . . . 6  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
5043, 9, 46, 47, 48, 49, 26ply1scltm 18088 . . . . 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 661 . . . 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 6338 . . 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 19052 . . . . . 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 2463 . . . . . . 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 6286 . . . . . . . . . . 11  |-  ( ( i  =  a  /\  j  =  b )  ->  ( i M j )  =  ( a M b ) )
6059fveq2d 5863 . . . . . . . . . 10  |-  ( ( i  =  a  /\  j  =  b )  ->  (coe1 `  ( i M j ) )  =  (coe1 `  ( a M b ) ) )
6160fveq1d 5861 . . . . . . . . 9  |-  ( ( i  =  a  /\  j  =  b )  ->  ( (coe1 `  ( i M j ) ) `  L )  =  ( (coe1 `  ( a M b ) ) `  L ) )
6261oveq1d 6292 . . . . . . . 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 466 . . . . . . 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 755 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  a  e.  N )
65 simprr 756 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  b  e.  N )
66 ovex 6302 . . . . . . . 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 6407 . . . . . 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 753 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  N  e.  Fin )
709ply1rng 18055 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
7170adantl 466 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  P  e.  Ring )
7271adantr 465 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  P  e.  Ring )
73 pm3.22 449 . . . . . . . . . . 11  |-  ( ( L  e.  NN0  /\  Q  e.  E )  ->  ( Q  e.  E  /\  L  e.  NN0 ) )
7473adantl 466 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( Q  e.  E  /\  L  e. 
NN0 ) )
75 eqid 2462 . . . . . . . . . . 11  |-  (coe1 `  Q
)  =  (coe1 `  Q
)
7675, 12, 9, 43coe1fvalcl 18017 . . . . . . . . . 10  |-  ( ( Q  e.  E  /\  L  e.  NN0 )  -> 
( (coe1 `  Q ) `  L )  e.  K
)
7774, 76syl 16 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( (coe1 `  Q ) `  L
)  e.  K )
789, 55, 43, 12ply1sclcl 18093 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
(coe1 `  Q ) `  L )  e.  K
)  ->  ( U `  ( (coe1 `  Q ) `  L ) )  e.  E )
793, 77, 78syl2anc 661 . . . . . . . 8  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( U `  ( (coe1 `  Q ) `  L ) )  e.  E )
8069, 72, 793jca 1171 . . . . . . 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 2462 . . . . . . . 8  |-  ( 0g
`  P )  =  ( 0g `  P
)
8210, 12, 81, 14, 13scmatscmide 18771 . . . . . . 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 471 . . . . . 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 2513 . . . . 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 2880 . . . 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 10801 . . . . . . . 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 18079 . . . . . . 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 1223 . . . . . 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 18687 . . . . 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 18965 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( U `  ( (coe1 `  Q ) `  L
) )  e.  E
)  ->  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  )  e.  B )
9269, 3, 79, 91syl3anc 1223 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  )  e.  B )
9310, 11eqmat 18688 . . . . 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 661 . . . 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 232 . . 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 2503 . 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 2507 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 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108   ifcif 3934   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7508   0cc0 9483   NN0cn0 10786   Basecbs 14481   .scvsca 14550   0gc0g 14686  .gcmg 15722  mulGrpcmgp 16926   1rcur 16938   Ringcrg 16981  algSccascl 17726  var1cv1 17981  Poly1cpl1 17982  coe1cco1 17983   Mat cmat 18671   matToPolyMat cmat2pmat 18967   decompPMat cdecpmat 19025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-ot 4031  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  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 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-ofr 6518  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-fzo 11784  df-seq 12066  df-hash 12363  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-hom 14570  df-cco 14571  df-0g 14688  df-gsum 14689  df-prds 14694  df-pws 14696  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-mhm 15772  df-submnd 15773  df-grp 15853  df-minusg 15854  df-sbg 15855  df-mulg 15856  df-subg 15988  df-ghm 16055  df-cntz 16145  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-subrg 17205  df-lmod 17292  df-lss 17357  df-sra 17596  df-rgmod 17597  df-ascl 17729  df-psr 17771  df-mvr 17772  df-mpl 17773  df-opsr 17775  df-psr1 17985  df-vr1 17986  df-ply1 17987  df-coe1 17988  df-dsmm 18525  df-frlm 18540  df-mamu 18648  df-mat 18672  df-mat2pmat 18970  df-decpmat 19026
This theorem is referenced by:  pmatcollpwscmat  19054
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