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Theorem cpmatel 19081
Description: Property of a constant polynomial matrix. (Contributed by AV, 15-Nov-2019.)
Hypotheses
Ref Expression
cpmat.s  |-  S  =  ( N ConstPolyMat  R )
cpmat.p  |-  P  =  (Poly1 `  R )
cpmat.c  |-  C  =  ( N Mat  P )
cpmat.b  |-  B  =  ( Base `  C
)
Assertion
Ref Expression
cpmatel  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( M  e.  S  <->  A. i  e.  N  A. j  e.  N  A. k  e.  NN  (
(coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) )
Distinct variable groups:    i, N, j, k    R, i, j, k    i, M, j, k
Allowed substitution hints:    B( i, j, k)    C( i, j, k)    P( i, j, k)    S( i, j, k)    V( i, j, k)

Proof of Theorem cpmatel
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 cpmat.s . . . . . 6  |-  S  =  ( N ConstPolyMat  R )
2 cpmat.p . . . . . 6  |-  P  =  (Poly1 `  R )
3 cpmat.c . . . . . 6  |-  C  =  ( N Mat  P )
4 cpmat.b . . . . . 6  |-  B  =  ( Base `  C
)
51, 2, 3, 4cpmat 19079 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  S  =  { m  e.  B  |  A. i  e.  N  A. j  e.  N  A. k  e.  NN  (
(coe1 `  ( i m j ) ) `  k )  =  ( 0g `  R ) } )
653adant3 1016 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  S  =  { m  e.  B  |  A. i  e.  N  A. j  e.  N  A. k  e.  NN  (
(coe1 `  ( i m j ) ) `  k )  =  ( 0g `  R ) } )
76eleq2d 2537 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( M  e.  S  <->  M  e.  { m  e.  B  |  A. i  e.  N  A. j  e.  N  A. k  e.  NN  ( (coe1 `  (
i m j ) ) `  k )  =  ( 0g `  R ) } ) )
8 oveq 6301 . . . . . . . . . 10  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
98fveq2d 5876 . . . . . . . . 9  |-  ( m  =  M  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
109fveq1d 5874 . . . . . . . 8  |-  ( m  =  M  ->  (
(coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  k ) )
1110eqeq1d 2469 . . . . . . 7  |-  ( m  =  M  ->  (
( (coe1 `  ( i m j ) ) `  k )  =  ( 0g `  R )  <-> 
( (coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) )
1211ralbidv 2906 . . . . . 6  |-  ( m  =  M  ->  ( A. k  e.  NN  ( (coe1 `  ( i m j ) ) `  k )  =  ( 0g `  R )  <->  A. k  e.  NN  ( (coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) )
1312ralbidv 2906 . . . . 5  |-  ( m  =  M  ->  ( A. j  e.  N  A. k  e.  NN  ( (coe1 `  ( i m j ) ) `  k )  =  ( 0g `  R )  <->  A. j  e.  N  A. k  e.  NN  ( (coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) )
1413ralbidv 2906 . . . 4  |-  ( m  =  M  ->  ( A. i  e.  N  A. j  e.  N  A. k  e.  NN  ( (coe1 `  ( i m j ) ) `  k )  =  ( 0g `  R )  <->  A. i  e.  N  A. j  e.  N  A. k  e.  NN  ( (coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) )
1514elrab 3266 . . 3  |-  ( M  e.  { m  e.  B  |  A. i  e.  N  A. j  e.  N  A. k  e.  NN  ( (coe1 `  (
i m j ) ) `  k )  =  ( 0g `  R ) }  <->  ( M  e.  B  /\  A. i  e.  N  A. j  e.  N  A. k  e.  NN  ( (coe1 `  (
i M j ) ) `  k )  =  ( 0g `  R ) ) )
167, 15syl6bb 261 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( M  e.  S  <->  ( M  e.  B  /\  A. i  e.  N  A. j  e.  N  A. k  e.  NN  (
(coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) ) )
17163anibar 1164 1  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( M  e.  S  <->  A. i  e.  N  A. j  e.  N  A. k  e.  NN  (
(coe1 `  ( i M j ) ) `  k )  =  ( 0g `  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   ` cfv 5594  (class class class)co 6295   Fincfn 7528   NNcn 10548   Basecbs 14507   0gc0g 14712  Poly1cpl1 18086  coe1cco1 18087   Mat cmat 18778   ConstPolyMat ccpmat 19073
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-cpmat 19076
This theorem is referenced by:  cpmatelimp  19082  cpmatel2  19083  1elcpmat  19085  cpmatmcl  19089  m2cpm  19111
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