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Theorem pmatcoe1fsupp 19329
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 3581 . . . . . 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 3723 . . . 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 7809 . . . . . . 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 7615 . . . . . . . 8  |-  { (coe1 `  ( M `  u
) ) }  e.  Fin
98a1i 11 . . . . . . 7  |-  ( ( N  e.  Fin  /\  u  e.  ( N  X.  N ) )  ->  { (coe1 `  ( M `  u ) ) }  e.  Fin )
109ralrimiva 2871 . . . . . 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 1017 . . . 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 7826 . . . 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 7762 . . . 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 5882 . . . . 5  |-  ( 0g
`  R )  e. 
_V
1816, 17eqeltri 2541 . . . 4  |-  .0.  e.  _V
1918a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  .0.  e.  _V )
20 elin 3683 . . . . . 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 4459 . . . . . . . . 9  |-  ( v  =  w  ->  (
v finSupp  .0.  <->  w finSupp  .0.  ) )
2221elrab 3257 . . . . . . . 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 2817 . . . 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 12102 . . . 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 1231 . 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 5040 . . . . . . . . . . . . . . . 16  |-  ( ( i  e.  N  /\  j  e.  N )  -> 
<. i ,  j >.  e.  ( N  X.  N
) )
32 fveq2 5872 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =  <. i ,  j
>.  ->  ( M `  u )  =  ( M `  <. i ,  j >. )
)
3332adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( M `  u )  =  ( M `  <. i ,  j >. )
)
3433fveq2d 5876 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  (coe1 `  ( M `  u )
)  =  (coe1 `  ( M `  <. i ,  j >. ) ) )
3534sneqd 4044 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  u ) ) }  =  {
(coe1 `  ( M `  <. i ,  j >.
) ) } )
3635eleq2d 2527 . . . . . . . . . . . . . . . . 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 6299 . . . . . . . . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  ( M `  <. i ,  j
>. )  =  (
i M j ) )
4039fveq2d 5876 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  (coe1 `  ( M `  <. i ,  j >. ) )  =  (coe1 `  ( i M j ) ) )
4140sneqd 4044 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  <. i ,  j >. )
) }  =  {
(coe1 `  ( i M j ) ) } )
4241eleq2d 2527 . . . . . . . . . . . . . . . . 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 253 . . . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . . . . . . 18  |-  (coe1 `  (
i M j ) )  e.  _V
4544snid 4060 . . . . . . . . . . . . . . . . 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 3215 . . . . . . . . . . . . . . 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 4337 . . . . . . . . . . . . . 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 pmatcoe1fsupp.c . . . . . . . . . . . . . . . 16  |-  C  =  ( N Mat  P )
52 eqid 2457 . . . . . . . . . . . . . . . 16  |-  ( Base `  P )  =  (
Base `  P )
53 pmatcoe1fsupp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  C
)
54 simprl 756 . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 2535 . . . . . . . . . . . . . . . . . . . 20  |-  ( M  e.  B  <->  M  e.  ( Base `  C )
)
5756biimpi 194 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  B  ->  M  e.  ( Base `  C
) )
58573ad2ant3 1019 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  M  e.  ( Base `  C
) )
5958ad3antrrr 729 . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . 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 19055 . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . 16  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
63 pmatcoe1fsupp.p . . . . . . . . . . . . . . . 16  |-  P  =  (Poly1 `  R )
64 eqid 2457 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
65 eqid 2457 . . . . . . . . . . . . . . . 16  |-  ( Base `  R )  =  (
Base `  R )
6662, 52, 63, 64, 65coe1fsupp 18381 . . . . . . . . . . . . . . 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 4468 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
v finSupp  .0.  <->  v finSupp  ( 0g `  R ) ) )
7069rabbidv 3101 . . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . . . 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 3684 . . . . . . . . . . . 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 755 . . . . . . . . . . . 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 5871 . . . . . . . . . . . . . . 15  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( w `  z
)  =  ( (coe1 `  ( i M j ) ) `  z
) )
7776eqeq1d 2459 . . . . . . . . . . . . . 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 4460 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
s  <  z  <->  s  <  x ) )
80 fveq2 5872 . . . . . . . . . . . . . . 15  |-  ( z  =  x  ->  (
(coe1 `  ( i M j ) ) `  z )  =  ( (coe1 `  ( i M j ) ) `  x ) )
8180eqeq1d 2459 . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . 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 2880 . . . . 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 2871 . . . 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 2932 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    i^i cin 3470    C_ wss 3471   {csn 4032   <.cop 4038   U_ciun 4332   class class class wbr 4456    X. cxp 5006   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   Fincfn 7535   finSupp cfsupp 7847    < clt 9645   NN0cn0 10816   Basecbs 14644   0gc0g 14857   Ringcrg 17325  Poly1cpl1 18343  coe1cco1 18344   Mat cmat 19036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-0g 14859  df-prds 14865  df-pws 14867  df-sra 17945  df-rgmod 17946  df-psr 18132  df-mpl 18134  df-opsr 18136  df-psr1 18346  df-ply1 18348  df-coe1 18349  df-dsmm 18890  df-frlm 18905  df-mat 19037
This theorem is referenced by:  decpmataa0  19396  decpmatmulsumfsupp  19401  pmatcollpw2lem  19405  pm2mpmhmlem1  19446
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