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Theorem pmatcoe1fsupp 19723
Description: For a polynomial matrix there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)
Hypotheses
Ref Expression
pmatcoe1fsupp.p  |-  P  =  (Poly1 `  R )
pmatcoe1fsupp.c  |-  C  =  ( N Mat  P )
pmatcoe1fsupp.b  |-  B  =  ( Base `  C
)
pmatcoe1fsupp.0  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
pmatcoe1fsupp  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  A. i  e.  N  A. j  e.  N  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
)
Distinct variable groups:    B, i,
j, x, s    x, C    i, M, j, x, s    i, N, j, x, s    R, i, j, s, x    .0. , i, j, s, x
Allowed substitution hints:    C( i, j, s)    P( x, i, j, s)

Proof of Theorem pmatcoe1fsupp
Dummy variables  v  u  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3546 . . . . . 6  |-  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  .0.  }  C_  ( ( Base `  R
)  ^m  NN0 )
21a1i 11 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  .0.  }  C_  ( ( Base `  R
)  ^m  NN0 ) )
32olcd 394 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) } 
C_  ( ( Base `  R )  ^m  NN0 )  \/  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  C_  (
( Base `  R )  ^m  NN0 ) ) )
4 inss 3691 . . . 4  |-  ( (
U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) } 
C_  ( ( Base `  R )  ^m  NN0 )  \/  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  C_  (
( Base `  R )  ^m  NN0 ) )  -> 
( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } )  C_  ( ( Base `  R
)  ^m  NN0 ) )
53, 4syl 17 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } )  C_  ( ( Base `  R
)  ^m  NN0 ) )
6 xpfi 7851 . . . . . . 7  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( N  X.  N
)  e.  Fin )
76anidms 649 . . . . . 6  |-  ( N  e.  Fin  ->  ( N  X.  N )  e. 
Fin )
8 snfi 7660 . . . . . . . 8  |-  { (coe1 `  ( M `  u
) ) }  e.  Fin
98a1i 11 . . . . . . 7  |-  ( ( N  e.  Fin  /\  u  e.  ( N  X.  N ) )  ->  { (coe1 `  ( M `  u ) ) }  e.  Fin )
109ralrimiva 2836 . . . . . 6  |-  ( N  e.  Fin  ->  A. u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  e.  Fin )
117, 10jca 534 . . . . 5  |-  ( N  e.  Fin  ->  (
( N  X.  N
)  e.  Fin  /\  A. u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  e.  Fin ) )
12113ad2ant1 1026 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
( N  X.  N
)  e.  Fin  /\  A. u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  e.  Fin ) )
13 iunfi 7871 . . . 4  |-  ( ( ( N  X.  N
)  e.  Fin  /\  A. u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  e.  Fin )  ->  U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  e.  Fin )
14 infi 7804 . . . 4  |-  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  e.  Fin  ->  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } )  e.  Fin )
1512, 13, 143syl 18 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } )  e.  Fin )
16 pmatcoe1fsupp.0 . . . . 5  |-  .0.  =  ( 0g `  R )
17 fvex 5891 . . . . 5  |-  ( 0g
`  R )  e. 
_V
1816, 17eqeltri 2503 . . . 4  |-  .0.  e.  _V
1918a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  .0.  e.  _V )
20 elin 3649 . . . . . 6  |-  ( w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } )  <->  ( w  e.  U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  /\  w  e.  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) )
21 breq1 4426 . . . . . . . . 9  |-  ( v  =  w  ->  (
v finSupp  .0.  <->  w finSupp  .0.  ) )
2221elrab 3228 . . . . . . . 8  |-  ( w  e.  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  <->  ( w  e.  ( ( Base `  R
)  ^m  NN0 )  /\  w finSupp  .0.  ) )
2322simprbi 465 . . . . . . 7  |-  ( w  e.  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  ->  w finSupp  .0.  )
2423adantl 467 . . . . . 6  |-  ( ( w  e.  U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  /\  w  e.  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } )  ->  w finSupp  .0.  )
2520, 24sylbi 198 . . . . 5  |-  ( w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } )  ->  w finSupp  .0.  )
2625rgen 2781 . . . 4  |-  A. w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) w finSupp  .0.
2726a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  A. w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) w finSupp  .0.  )
28 fsuppmapnn0fiub0 12211 . . . 4  |-  ( ( ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } )  C_  (
( Base `  R )  ^m  NN0 )  /\  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } )  e.  Fin  /\  .0.  e.  _V )  ->  ( A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) w finSupp  .0.  ->  E. s  e.  NN0  A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) ) )
2928imp 430 . . 3  |-  ( ( ( ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } )  C_  (
( Base `  R )  ^m  NN0 )  /\  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } )  e.  Fin  /\  .0.  e.  _V )  /\  A. w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) w finSupp  .0.  )  ->  E. s  e.  NN0  A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )
305, 15, 19, 27, 29syl31anc 1267 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  E. s  e.  NN0  A. w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )
31 opelxpi 4885 . . . . . . . . . . . . . . . 16  |-  ( ( i  e.  N  /\  j  e.  N )  -> 
<. i ,  j >.  e.  ( N  X.  N
) )
32 fveq2 5881 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. i ,  j
>.  ->  ( M `  u )  =  ( M `  <. i ,  j >. )
)
3332adantl 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( M `  u )  =  ( M `  <. i ,  j >. )
)
3433fveq2d 5885 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  (coe1 `  ( M `  u )
)  =  (coe1 `  ( M `  <. i ,  j >. ) ) )
3534sneqd 4010 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  u ) ) }  =  {
(coe1 `  ( M `  <. i ,  j >.
) ) } )
3635eleq2d 2492 . . . . . . . . . . . . . . . . 17  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( (coe1 `  ( i M j ) )  e.  {
(coe1 `  ( M `  u ) ) }  <-> 
(coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( M `  <. i ,  j >.
) ) } ) )
37 df-ov 6308 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i M j )  =  ( M `  <. i ,  j >. )
3837a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( i M j )  =  ( M `  <. i ,  j >. )
)
3938eqcomd 2430 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( M `  <. i ,  j
>. )  =  (
i M j ) )
4039fveq2d 5885 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  (coe1 `  ( M `  <. i ,  j >. ) )  =  (coe1 `  ( i M j ) ) )
4140sneqd 4010 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  <. i ,  j >. )
) }  =  {
(coe1 `  ( i M j ) ) } )
4241eleq2d 2492 . . . . . . . . . . . . . . . . 17  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( (coe1 `  ( i M j ) )  e.  {
(coe1 `  ( M `  <. i ,  j >.
) ) }  <->  (coe1 `  (
i M j ) )  e.  { (coe1 `  ( i M j ) ) } ) )
4336, 42bitrd 256 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( (coe1 `  ( i M j ) )  e.  {
(coe1 `  ( M `  u ) ) }  <-> 
(coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( i M j ) ) } ) )
44 fvex 5891 . . . . . . . . . . . . . . . . . 18  |-  (coe1 `  (
i M j ) )  e.  _V
4544snid 4026 . . . . . . . . . . . . . . . . 17  |-  (coe1 `  (
i M j ) )  e.  { (coe1 `  ( i M j ) ) }
4645a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( i  e.  N  /\  j  e.  N )  ->  (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( i M j ) ) } )
4731, 43, 46rspcedvd 3187 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  N  /\  j  e.  N )  ->  E. u  e.  ( N  X.  N ) (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( M `  u ) ) } )
4847adantl 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  E. u  e.  ( N  X.  N
) (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( M `  u ) ) } )
49 eliun 4304 . . . . . . . . . . . . . 14  |-  ( (coe1 `  ( i M j ) )  e.  U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  <->  E. u  e.  ( N  X.  N
) (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( M `  u ) ) } )
5048, 49sylibr 215 . . . . . . . . . . . . 13  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  (coe1 `  (
i M j ) )  e.  U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) } )
51 pmatcoe1fsupp.c . . . . . . . . . . . . . . . 16  |-  C  =  ( N Mat  P )
52 eqid 2422 . . . . . . . . . . . . . . . 16  |-  ( Base `  P )  =  (
Base `  P )
53 pmatcoe1fsupp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  C
)
54 simprl 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  i  e.  N )
55 simprr 764 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  j  e.  N )
5653eleq2i 2499 . . . . . . . . . . . . . . . . . . . 20  |-  ( M  e.  B  <->  M  e.  ( Base `  C )
)
5756biimpi 197 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  B  ->  M  e.  ( Base `  C
) )
58573ad2ant3 1028 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  M  e.  ( Base `  C
) )
5958ad3antrrr 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  M  e.  ( Base `  C )
)
6059, 56sylibr 215 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  M  e.  B )
6151, 52, 53, 54, 55, 60matecld 19449 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  ( i M j )  e.  ( Base `  P
) )
62 eqid 2422 . . . . . . . . . . . . . . . 16  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
63 pmatcoe1fsupp.p . . . . . . . . . . . . . . . 16  |-  P  =  (Poly1 `  R )
64 eqid 2422 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
65 eqid 2422 . . . . . . . . . . . . . . . 16  |-  ( Base `  R )  =  (
Base `  R )
6662, 52, 63, 64, 65coe1fsupp 18806 . . . . . . . . . . . . . . 15  |-  ( ( i M j )  e.  ( Base `  P
)  ->  (coe1 `  (
i M j ) )  e.  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  ( 0g
`  R ) } )
6761, 66syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  (coe1 `  (
i M j ) )  e.  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  ( 0g
`  R ) } )
6816a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  .0.  =  ( 0g `  R ) )
6968breq2d 4435 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
v finSupp  .0.  <->  v finSupp  ( 0g `  R ) ) )
7069rabbidv 3071 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  .0.  }  =  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  ( 0g `  R ) } )
7170eleq2d 2492 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  ( i M j ) )  e. 
{ v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  <->  (coe1 `  (
i M j ) )  e.  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  ( 0g
`  R ) } ) )
7271ad3antrrr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  ( (coe1 `  ( i M j ) )  e.  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  }  <->  (coe1 `  ( i M j ) )  e. 
{ v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  ( 0g `  R ) } ) )
7367, 72mpbird 235 . . . . . . . . . . . . 13  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  (coe1 `  (
i M j ) )  e.  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  .0.  }
)
7450, 73elind 3650 . . . . . . . . . . . 12  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  (coe1 `  (
i M j ) )  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) )
75 simplr 760 . . . . . . . . . . . 12  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  x  e.  NN0 )
76 fveq1 5880 . . . . . . . . . . . . . . 15  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( w `  z
)  =  ( (coe1 `  ( i M j ) ) `  z
) )
7776eqeq1d 2424 . . . . . . . . . . . . . 14  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( ( w `  z )  =  .0.  <->  ( (coe1 `  ( i M j ) ) `  z )  =  .0.  ) )
7877imbi2d 317 . . . . . . . . . . . . 13  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( ( s  < 
z  ->  ( w `  z )  =  .0.  )  <->  ( s  < 
z  ->  ( (coe1 `  ( i M j ) ) `  z
)  =  .0.  )
) )
79 breq2 4427 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
s  <  z  <->  s  <  x ) )
80 fveq2 5881 . . . . . . . . . . . . . . 15  |-  ( z  =  x  ->  (
(coe1 `  ( i M j ) ) `  z )  =  ( (coe1 `  ( i M j ) ) `  x ) )
8180eqeq1d 2424 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
( (coe1 `  ( i M j ) ) `  z )  =  .0.  <->  ( (coe1 `  ( i M j ) ) `  x )  =  .0.  ) )
8279, 81imbi12d 321 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  (
( s  <  z  ->  ( (coe1 `  ( i M j ) ) `  z )  =  .0.  )  <->  ( s  < 
x  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) )
8378, 82rspc2v 3191 . . . . . . . . . . . 12  |-  ( ( (coe1 `  ( i M j ) )  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } )  /\  x  e.  NN0 )  ->  ( A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  <  z  ->  ( w `  z
)  =  .0.  )  ->  ( s  <  x  ->  ( (coe1 `  ( i M j ) ) `  x )  =  .0.  ) ) )
8474, 75, 83syl2anc 665 . . . . . . . . . . 11  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  (
i  e.  N  /\  j  e.  N )
)  ->  ( A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  )  ->  ( s  <  x  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) )
8584ex 435 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  M  e.  B
)  /\  s  e.  NN0 )  /\  x  e. 
NN0 )  ->  (
( i  e.  N  /\  j  e.  N
)  ->  ( A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  )  ->  ( s  <  x  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) ) )
8685com23 81 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  M  e.  B
)  /\  s  e.  NN0 )  /\  x  e. 
NN0 )  ->  ( A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  <  z  ->  ( w `  z
)  =  .0.  )  ->  ( ( i  e.  N  /\  j  e.  N )  ->  (
s  <  x  ->  ( (coe1 `  ( i M j ) ) `  x )  =  .0.  ) ) ) )
8786impancom 441 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  M  e.  B
)  /\  s  e.  NN0 )  /\  A. w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )  ->  (
x  e.  NN0  ->  ( ( i  e.  N  /\  j  e.  N
)  ->  ( s  <  x  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) ) )
8887imp 430 . . . . . . 7  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )  /\  x  e.  NN0 )  ->  (
( i  e.  N  /\  j  e.  N
)  ->  ( s  <  x  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) )
8988com23 81 . . . . . 6  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )  /\  x  e.  NN0 )  ->  (
s  <  x  ->  ( ( i  e.  N  /\  j  e.  N
)  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) )
9089ralrimdvv 2845 . . . . 5  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  /\  A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )  /\  x  e.  NN0 )  ->  (
s  <  x  ->  A. i  e.  N  A. j  e.  N  (
(coe1 `  ( i M j ) ) `  x )  =  .0.  ) )
9190ralrimiva 2836 . . . 4  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  M  e.  B
)  /\  s  e.  NN0 )  /\  A. w  e.  ( U_ u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  i^i  {
v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  ) )  ->  A. x  e.  NN0  ( s  < 
x  ->  A. i  e.  N  A. j  e.  N  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
)
9291ex 435 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  s  e.  NN0 )  ->  ( A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  i^i  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  <  z  ->  ( w `  z
)  =  .0.  )  ->  A. x  e.  NN0  ( s  <  x  ->  A. i  e.  N  A. j  e.  N  ( (coe1 `  ( i M j ) ) `  x )  =  .0.  ) ) )
9392reximdva 2897 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  ( E. s  e.  NN0  A. w  e.  ( U_ u  e.  ( N  X.  N ) { (coe1 `  ( M `  u
) ) }  i^i  { v  e.  ( (
Base `  R )  ^m  NN0 )  |  v finSupp  .0.  } ) A. z  e.  NN0  ( s  < 
z  ->  ( w `  z )  =  .0.  )  ->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  A. i  e.  N  A. j  e.  N  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) )
9430, 93mpd 15 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  A. i  e.  N  A. j  e.  N  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772   {crab 2775   _Vcvv 3080    i^i cin 3435    C_ wss 3436   {csn 3998   <.cop 4004   U_ciun 4299   class class class wbr 4423    X. cxp 4851   ` cfv 5601  (class class class)co 6305    ^m cmap 7483   Fincfn 7580   finSupp cfsupp 7892    < clt 9682   NN0cn0 10876   Basecbs 15120   0gc0g 15337   Ringcrg 17779  Poly1cpl1 18769  coe1cco1 18770   Mat cmat 19430
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 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
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-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-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-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-hom 15213  df-cco 15214  df-0g 15339  df-prds 15345  df-pws 15347  df-sra 18394  df-rgmod 18395  df-psr 18579  df-mpl 18581  df-opsr 18583  df-psr1 18772  df-ply1 18774  df-coe1 18775  df-dsmm 19293  df-frlm 19308  df-mat 19431
This theorem is referenced by:  decpmataa0  19790  decpmatmulsumfsupp  19795  pmatcollpw2lem  19799  pm2mpmhmlem1  19840
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