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Theorem pmatcoe1fsupp 19071
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 3590 . . . . . 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 3732 . . . 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 7803 . . . . . . 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 7608 . . . . . . . 8  |-  { (coe1 `  ( M `  u
) ) }  e.  Fin
98a1i 11 . . . . . . 7  |-  ( ( N  e.  Fin  /\  u  e.  ( N  X.  N ) )  ->  { (coe1 `  ( M `  u ) ) }  e.  Fin )
109ralrimiva 2881 . . . . . 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 7820 . . . 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 7755 . . . 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 2551 . . . 4  |-  .0.  e.  _V
1918a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  .0.  e.  _V )
20 elin 3692 . . . . . 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 4456 . . . . . . . . 9  |-  ( v  =  w  ->  (
v finSupp  .0.  <->  w finSupp  .0.  ) )
2221elrab 3266 . . . . . . . 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 2827 . . . 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 12079 . . . 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 5037 . . . . . . . . . . . . . . . 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 4045 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  u ) ) }  =  {
(coe1 `  ( M `  <. i ,  j >.
) ) } )
3635eleq2d 2537 . . . . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . . . . . . 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 4045 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( i  e.  N  /\  j  e.  N
)  /\  u  =  <. i ,  j >.
)  ->  { (coe1 `  ( M `  <. i ,  j >. )
) }  =  {
(coe1 `  ( i M j ) ) } )
4241eleq2d 2537 . . . . . . . . . . . . . . . . 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 4061 . . . . . . . . . . . . . . . . 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 3224 . . . . . . . . . . . . . . 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 4336 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  ( Base `  P )  =  (
Base `  P )
53 pmatcoe1fsupp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  C
)
54 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 )
55 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 )
5653eleq2i 2545 . . . . . . . . . . . . . . . . . . . 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 18797 . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  (coe1 `  (
i M j ) )  =  (coe1 `  (
i M j ) )
63 pmatcoe1fsupp.p . . . . . . . . . . . . . . . 16  |-  P  =  (Poly1 `  R )
64 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
65 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( Base `  R )  =  (
Base `  R )
6662, 52, 63, 64, 65coe1fsupp 18124 . . . . . . . . . . . . . . 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 4465 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
v finSupp  .0.  <->  v finSupp  ( 0g `  R ) ) )
7069rabbidv 3110 . . . . . . . . . . . . . . . 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 2537 . . . . . . . . . . . . . . 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 3693 . . . . . . . . . . . 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 5871 . . . . . . . . . . . . . . 15  |-  ( w  =  (coe1 `  ( i M j ) )  -> 
( w `  z
)  =  ( (coe1 `  ( i M j ) ) `  z
) )
7776eqeq1d 2469 . . . . . . . . . . . . . 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 4457 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
s  <  z  <->  s  <  x ) )
80 fveq2 5872 . . . . . . . . . . . . . . 15  |-  ( z  =  x  ->  (
(coe1 `  ( i M j ) ) `  z )  =  ( (coe1 `  ( i M j ) ) `  x ) )
8180eqeq1d 2469 . . . . . . . . . . . . . 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 3228 . . . . . . . . . . . 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 2890 . . . . 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 2881 . . . 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 2942 . 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 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    i^i cin 3480    C_ wss 3481   {csn 4033   <.cop 4039   U_ciun 4331   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Fincfn 7528   finSupp cfsupp 7841    < clt 9640   NN0cn0 10807   Basecbs 14507   0gc0g 14712   Ringcrg 17070  Poly1cpl1 18086  coe1cco1 18087   Mat cmat 18778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-sra 17689  df-rgmod 17690  df-psr 17875  df-mpl 17877  df-opsr 17879  df-psr1 18089  df-ply1 18091  df-coe1 18092  df-dsmm 18632  df-frlm 18647  df-mat 18779
This theorem is referenced by:  decpmataa0  19138  decpmatmulsumfsupp  19143  pmatcollpw2lem  19147  pm2mpmhmlem1  19188
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