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Theorem pmatcollpwscmatlem2 19268
Description: Lemma 2 for pmatcollpwscmat 19269. (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 976 . . . . . . 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 19180 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  Q  e.  E )  ->  ( Q  .*  .1.  )  e.  B )
168, 15syl5eqel 2535 . . . . . 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 756 . . . . 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 19245 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B  /\  L  e. 
NN0 )  ->  ( M decompPMat  L )  e.  D
)
223, 17, 18, 21syl3anc 1229 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( M decompPMat  L )  e.  D )
23 df-3an 976 . . . 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 2443 . . . 4  |-  (algSc `  P )  =  (algSc `  P )
2725, 19, 20, 9, 26mat2pmatval 19202 . . 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 1177 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( R  e.  Ring  /\  M  e.  B  /\  L  e.  NN0 ) )
30293ad2ant1 1018 . . . . 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 996 . . . . 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 19244 . . . . 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 5860 . . 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 6346 . 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 1061 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
37 simp2 998 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
38 simp3 999 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  j  e.  N )
39173ad2ant1 1018 . . . . . . 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 18905 . . . . . 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 1018 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  i  e.  N  /\  j  e.  N )  ->  L  e.  NN0 )
42 eqid 2443 . . . . . . 7  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
43 pmatcollpwscmat.k . . . . . . 7  |-  K  =  ( Base `  R
)
4442, 12, 9, 43coe1fvalcl 18229 . . . . . 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 2443 . . . . . 6  |-  (var1 `  R
)  =  (var1 `  R
)
47 eqid 2443 . . . . . 6  |-  ( .s
`  P )  =  ( .s `  P
)
48 eqid 2443 . . . . . 6  |-  (mulGrp `  P )  =  (mulGrp `  P )
49 eqid 2443 . . . . . 6  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
5043, 9, 46, 47, 48, 49, 26ply1scltm 18300 . . . . 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 6346 . . 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 19267 . . . . . 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 2444 . . . . . . 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 6290 . . . . . . . . . . 11  |-  ( ( i  =  a  /\  j  =  b )  ->  ( i M j )  =  ( a M b ) )
6059fveq2d 5860 . . . . . . . . . 10  |-  ( ( i  =  a  /\  j  =  b )  ->  (coe1 `  ( i M j ) )  =  (coe1 `  ( a M b ) ) )
6160fveq1d 5858 . . . . . . . . 9  |-  ( ( i  =  a  /\  j  =  b )  ->  ( (coe1 `  ( i M j ) ) `  L )  =  ( (coe1 `  ( a M b ) ) `  L ) )
6261oveq1d 6296 . . . . . . . 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 756 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  a  e.  N )
65 simprr 757 . . . . . . 7  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  /\  ( a  e.  N  /\  b  e.  N
) )  ->  b  e.  N )
66 ovex 6309 . . . . . . . 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 6415 . . . . . 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 )
709ply1ring 18267 . . . . . . . . . 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 2443 . . . . . . . . . . 11  |-  (coe1 `  Q
)  =  (coe1 `  Q
)
7675, 12, 9, 43coe1fvalcl 18229 . . . . . . . . . 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 18305 . . . . . . . . 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 1177 . . . . . . 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 2443 . . . . . . . 8  |-  ( 0g
`  P )  =  ( 0g `  P
)
8210, 12, 81, 14, 13scmatscmide 18986 . . . . . . 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 2494 . . . . 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 2864 . . . 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 10817 . . . . . . . 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 18291 . . . . . . 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 1229 . . . . . 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 18902 . . . . 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 19180 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  ( U `  ( (coe1 `  Q ) `  L
) )  e.  E
)  ->  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  )  e.  B )
9269, 3, 79, 91syl3anc 1229 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( L  e.  NN0  /\  Q  e.  E ) )  ->  ( ( U `  ( (coe1 `  Q ) `  L
) )  .*  .1.  )  e.  B )
9310, 11eqmat 18903 . . . . 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 2484 . 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 2488 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095   ifcif 3926   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Fincfn 7518   0cc0 9495   NN0cn0 10802   Basecbs 14613   .scvsca 14682   0gc0g 14818  .gcmg 16034  mulGrpcmgp 17119   1rcur 17131   Ringcrg 17176  algSccascl 17938  var1cv1 18193  Poly1cpl1 18194  coe1cco1 18195   Mat cmat 18886   matToPolyMat cmat2pmat 19182   decompPMat cdecpmat 19240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-fzo 11806  df-seq 12089  df-hash 12387  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-hom 14702  df-cco 14703  df-0g 14820  df-gsum 14821  df-prds 14826  df-pws 14828  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-mhm 15944  df-submnd 15945  df-grp 16035  df-minusg 16036  df-sbg 16037  df-mulg 16038  df-subg 16176  df-ghm 16243  df-cntz 16333  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-ring 17178  df-subrg 17405  df-lmod 17492  df-lss 17557  df-sra 17796  df-rgmod 17797  df-ascl 17941  df-psr 17983  df-mvr 17984  df-mpl 17985  df-opsr 17987  df-psr1 18197  df-vr1 18198  df-ply1 18199  df-coe1 18200  df-dsmm 18740  df-frlm 18755  df-mamu 18863  df-mat 18887  df-mat2pmat 19185  df-decpmat 19241
This theorem is referenced by:  pmatcollpwscmat  19269
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