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Theorem pmatcoe1fsupp 30892
Description: For a matrix consisting of polynomials there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-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 3437 . . . . . 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 393 . . . 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 3579 . . . 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 16 . . 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 7583 . . . . . . 7  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( N  X.  N
)  e.  Fin )
76anidms 645 . . . . . 6  |-  ( N  e.  Fin  ->  ( N  X.  N )  e. 
Fin )
8 snfi 7390 . . . . . . . 8  |-  { (coe1 `  ( M `  u
) ) }  e.  Fin
98a1i 11 . . . . . . 7  |-  ( ( N  e.  Fin  /\  u  e.  ( N  X.  N ) )  ->  { (coe1 `  ( M `  u ) ) }  e.  Fin )
109ralrimiva 2799 . . . . . 6  |-  ( N  e.  Fin  ->  A. u  e.  ( N  X.  N
) { (coe1 `  ( M `  u )
) }  e.  Fin )
117, 10jca 532 . . . . 5  |-  ( N  e.  Fin  ->  (
( N  X.  N
)  e.  Fin  /\  A. u  e.  ( N  X.  N ) { (coe1 `  ( M `  u ) ) }  e.  Fin ) )
12113ad2ant1 1009 . . . 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 7599 . . . 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 7536 . . . 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 20 . . 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 5701 . . . . 5  |-  ( 0g
`  R )  e. 
_V
1816, 17eqeltri 2513 . . . 4  |-  .0.  e.  _V
1918a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  .0.  e.  _V )
20 elin 3539 . . . . . 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 4295 . . . . . . . . 9  |-  ( v  =  w  ->  (
v finSupp  .0.  <->  w finSupp  .0.  ) )
2221elrab 3117 . . . . . . . 8  |-  ( w  e.  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  <->  ( w  e.  ( ( Base `  R
)  ^m  NN0 )  /\  w finSupp  .0.  ) )
2322simprbi 464 . . . . . . 7  |-  ( w  e.  { v  e.  ( ( Base `  R
)  ^m  NN0 )  |  v finSupp  .0.  }  ->  w finSupp  .0.  )
2423adantl 466 . . . . . 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 195 . . . . 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 30801 . . . 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 429 . . 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 1221 . 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 fvex 5701 . . . . . . . . . . . . . . . . 17  |-  (coe1 `  (
i M j ) )  e.  _V
3231snid 3905 . . . . . . . . . . . . . . . 16  |-  (coe1 `  (
i M j ) )  e.  { (coe1 `  ( i M j ) ) }
33 opelxpi 4871 . . . . . . . . . . . . . . . . 17  |-  ( ( i  e.  N  /\  j  e.  N )  -> 
<. i ,  j >.  e.  ( N  X.  N
) )
34 fveq2 5691 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  <. i ,  j
>.  ->  ( M `  u )  =  ( M `  <. i ,  j >. )
)
3534adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( M `  u )  =  ( M `  <. i ,  j >. )
)
3635fveq2d 5695 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  (coe1 `  ( M `  u )
)  =  (coe1 `  ( M `  <. i ,  j >. ) ) )
3736sneqd 3889 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  u ) ) }  =  {
(coe1 `  ( M `  <. i ,  j >.
) ) } )
3837eleq2d 2510 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 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 >.
) ) } ) )
39 df-ov 6094 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i M j )  =  ( M `  <. i ,  j >. )
4039a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( i M j )  =  ( M `  <. i ,  j >. )
)
4140eqcomd 2448 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( M `  <. i ,  j
>. )  =  (
i M j ) )
4241fveq2d 5695 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  (coe1 `  ( M `  <. i ,  j >. ) )  =  (coe1 `  ( i M j ) ) )
4342sneqd 3889 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  <. i ,  j >. )
) }  =  {
(coe1 `  ( i M j ) ) } )
4443eleq2d 2510 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 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 ) ) } ) )
4538, 44bitrd 253 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 ) ) } ) )
4633, 45rspcedv 3077 . . . . . . . . . . . . . . . 16  |-  ( ( i  e.  N  /\  j  e.  N )  ->  ( (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( i M j ) ) }  ->  E. u  e.  ( N  X.  N ) (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( M `  u ) ) } ) )
4732, 46mpi 17 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  N  /\  j  e.  N )  ->  E. u  e.  ( N  X.  N ) (coe1 `  ( i M j ) )  e. 
{ (coe1 `  ( M `  u ) ) } )
4847adantl 466 . . . . . . . . . . . . . 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 4175 . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . 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 simprl 755 . . . . . . . . . . . . . . . 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 )
52 simprr 756 . . . . . . . . . . . . . . . 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 )
53 pmatcoe1fsupp.b . . . . . . . . . . . . . . . . . . . 20  |-  B  =  ( Base `  C
)
5453eleq2i 2507 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  B  <->  M  e.  ( Base `  C )
)
5554biimpi 194 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  B  ->  M  e.  ( Base `  C
) )
56553ad2ant3 1011 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  M  e.  ( Base `  C
) )
5756ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( 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 )
)
58 pmatcoe1fsupp.c . . . . . . . . . . . . . . . . 17  |-  C  =  ( N Mat  P )
59 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  ( Base `  P )  =  (
Base `  P )
6058, 59matecl 18326 . . . . . . . . . . . . . . . 16  |-  ( ( i  e.  N  /\  j  e.  N  /\  M  e.  ( Base `  C ) )  -> 
( i M j )  e.  ( Base `  P ) )
6151, 52, 57, 60syl3anc 1218 . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . 16  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
63 pmatcoe1fsupp.p . . . . . . . . . . . . . . . 16  |-  P  =  (Poly1 `  R )
64 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( Base `  R )  =  (
Base `  R )
65 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
6662, 59, 63, 64, 65coe1fsupp 30831 . . . . . . . . . . . . . . 15  |-  ( ( i M j )  e.  ( Base `  P
)  ->  (coe1 `  (
i M j ) )  e.  { v  e.  ( ( Base `  R )  ^m  NN0 )  |  v finSupp  ( 0g
`  R ) } )
6761, 66syl 16 . . . . . . . . . . . . . 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 4304 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
v finSupp  .0.  <->  v finSupp  ( 0g `  R ) ) )
7069rabbidv 2964 . . . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 3540 . . . . . . . . . . . 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 754 . . . . . . . . . . . 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 5690 . . . . . . . . . . . . . . 15  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( w `  z
)  =  ( (coe1 `  ( i M j ) ) `  z
) )
7776eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( ( w `  z )  =  .0.  <->  ( (coe1 `  ( i M j ) ) `  z )  =  .0.  ) )
7877imbi2d 316 . . . . . . . . . . . . 13  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( ( s  < 
z  ->  ( w `  z )  =  .0.  )  <->  ( s  < 
z  ->  ( (coe1 `  ( i M j ) ) `  z
)  =  .0.  )
) )
79 breq2 4296 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
s  <  z  <->  s  <  x ) )
80 fveq2 5691 . . . . . . . . . . . . . . 15  |-  ( z  =  x  ->  (
(coe1 `  ( i M j ) ) `  z )  =  ( (coe1 `  ( i M j ) ) `  x ) )
8180eqeq1d 2451 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
( (coe1 `  ( i M j ) ) `  z )  =  .0.  <->  ( (coe1 `  ( i M j ) ) `  x )  =  .0.  ) )
8279, 81imbi12d 320 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  (
( s  <  z  ->  ( (coe1 `  ( i M j ) ) `  z )  =  .0.  )  <->  ( s  < 
x  ->  ( (coe1 `  ( i M j ) ) `  x
)  =  .0.  )
) )
8378, 82rspc2v 3079 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 434 . . . . . . . . . 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 78 . . . . . . . . 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 440 . . . . . . . 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 429 . . . . . . 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 78 . . . . . 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 2810 . . . . 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 2799 . . . 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 434 . . 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 2828 . 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 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972    i^i cin 3327    C_ wss 3328   {csn 3877   <.cop 3883   U_ciun 4171   class class class wbr 4292    X. cxp 4838   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   Fincfn 7310   finSupp cfsupp 7620    < clt 9418   NN0cn0 10579   Basecbs 14174   0gc0g 14378   Ringcrg 16645  Poly1cpl1 17633  coe1cco1 17634   Mat cmat 18280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-prds 14386  df-pws 14388  df-sra 17253  df-rgmod 17254  df-psr 17423  df-mpl 17425  df-opsr 17427  df-psr1 17636  df-ply1 17638  df-coe1 17639  df-dsmm 18157  df-frlm 18172  df-mat 18282
This theorem is referenced by:  pmatcollpw2lem  30905  pmattomply1lem  30908  pmattomply1mhmlem0  30927  pmattomply1mhmlem1  30928
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