MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptcoe1matfsupp Structured version   Unicode version

Theorem mptcoe1matfsupp 19172
Description: The mapping extracting the entries of the coefficient matrices of a polynomial over matrices at a fixed position is finitely supported. (Contributed by AV, 6-Oct-2019.) (Proof shortened by AV, 23-Dec-2019.)
Hypotheses
Ref Expression
mptcoe1matfsupp.a  |-  A  =  ( N Mat  R )
mptcoe1matfsupp.q  |-  Q  =  (Poly1 `  A )
mptcoe1matfsupp.l  |-  L  =  ( Base `  Q
)
Assertion
Ref Expression
mptcoe1matfsupp  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (
k  e.  NN0  |->  ( I ( (coe1 `  O ) `  k ) J ) ) finSupp  ( 0g `  R ) )
Distinct variable groups:    k, L    k, I    k, J    k, N    k, O    R, k
Allowed substitution hints:    A( k)    Q( k)

Proof of Theorem mptcoe1matfsupp
Dummy variables  c 
s  x  i  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5882 . . 3  |-  ( 0g
`  R )  e. 
_V
21a1i 11 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  ( 0g `  R )  e. 
_V )
3 mptcoe1matfsupp.a . . 3  |-  A  =  ( N Mat  R )
4 eqid 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2467 . . 3  |-  ( Base `  A )  =  (
Base `  A )
6 simp2 997 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  I  e.  N )
76adantr 465 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  I  e.  N )
8 simp3 998 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  J  e.  N )
98adantr 465 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  J  e.  N )
10 simp3 998 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  O  e.  L )
11103ad2ant1 1017 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  O  e.  L )
12 eqid 2467 . . . . 5  |-  (coe1 `  O
)  =  (coe1 `  O
)
13 mptcoe1matfsupp.l . . . . 5  |-  L  =  ( Base `  Q
)
14 mptcoe1matfsupp.q . . . . 5  |-  Q  =  (Poly1 `  A )
1512, 13, 14, 5coe1fvalcl 18121 . . . 4  |-  ( ( O  e.  L  /\  k  e.  NN0 )  -> 
( (coe1 `  O ) `  k )  e.  (
Base `  A )
)
1611, 15sylan 471 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  ( (coe1 `  O ) `  k
)  e.  ( Base `  A ) )
173, 4, 5, 7, 9, 16matecld 18797 . 2  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  ( I
( (coe1 `  O ) `  k ) J )  e.  ( Base `  R
) )
18 eqid 2467 . . . . . . 7  |-  ( 0g
`  A )  =  ( 0g `  A
)
1912, 13, 14, 18, 5coe1fsupp 18124 . . . . . 6  |-  ( O  e.  L  ->  (coe1 `  O )  e.  {
c  e.  ( (
Base `  A )  ^m  NN0 )  |  c finSupp 
( 0g `  A
) } )
20 elrabi 3263 . . . . . 6  |-  ( (coe1 `  O )  e.  {
c  e.  ( (
Base `  A )  ^m  NN0 )  |  c finSupp 
( 0g `  A
) }  ->  (coe1 `  O )  e.  ( ( Base `  A
)  ^m  NN0 ) )
2111, 19, 203syl 20 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (coe1 `  O )  e.  ( ( Base `  A
)  ^m  NN0 ) )
22 fvex 5882 . . . . 5  |-  ( 0g
`  A )  e. 
_V
2321, 22jctir 538 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (
(coe1 `  O )  e.  ( ( Base `  A
)  ^m  NN0 )  /\  ( 0g `  A )  e.  _V ) )
2412, 13, 14, 18coe1sfi 18122 . . . . 5  |-  ( O  e.  L  ->  (coe1 `  O ) finSupp  ( 0g `  A ) )
2511, 24syl 16 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (coe1 `  O ) finSupp  ( 0g `  A ) )
26 fsuppmapnn0ub 12081 . . . 4  |-  ( ( (coe1 `  O )  e.  ( ( Base `  A
)  ^m  NN0 )  /\  ( 0g `  A )  e.  _V )  -> 
( (coe1 `  O ) finSupp  ( 0g `  A )  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  ( (coe1 `  O ) `  x )  =  ( 0g `  A ) ) ) )
2723, 25, 26sylc 60 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  ( (coe1 `  O ) `  x
)  =  ( 0g
`  A ) ) )
28 csbov 6327 . . . . . . . . . 10  |-  [_ x  /  k ]_ (
I ( (coe1 `  O
) `  k ) J )  =  ( I [_ x  / 
k ]_ ( (coe1 `  O
) `  k ) J )
29 csbfv 5910 . . . . . . . . . . 11  |-  [_ x  /  k ]_ (
(coe1 `  O ) `  k )  =  ( (coe1 `  O ) `  x )
3029oveqi 6308 . . . . . . . . . 10  |-  ( I
[_ x  /  k ]_ ( (coe1 `  O ) `  k ) J )  =  ( I ( (coe1 `  O ) `  x ) J )
3128, 30eqtri 2496 . . . . . . . . 9  |-  [_ x  /  k ]_ (
I ( (coe1 `  O
) `  k ) J )  =  ( I ( (coe1 `  O
) `  x ) J )
3231a1i 11 . . . . . . . 8  |-  ( ( ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  s  <  x )  /\  (
(coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  [_ x  / 
k ]_ ( I ( (coe1 `  O ) `  k ) J )  =  ( I ( (coe1 `  O ) `  x ) J ) )
33 oveq 6301 . . . . . . . . 9  |-  ( ( (coe1 `  O ) `  x )  =  ( 0g `  A )  ->  ( I ( (coe1 `  O ) `  x ) J )  =  ( I ( 0g `  A ) J ) )
3433adantl 466 . . . . . . . 8  |-  ( ( ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  s  <  x )  /\  (
(coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  ( I
( (coe1 `  O ) `  x ) J )  =  ( I ( 0g `  A ) J ) )
35 eqid 2467 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
363, 35mat0op 18790 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( i  e.  N ,  j  e.  N  |->  ( 0g
`  R ) ) )
37363adant3 1016 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( 0g `  A )  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R
) ) )
38373ad2ant1 1017 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  ( 0g `  A )  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R
) ) )
39 eqidd 2468 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  ( i  =  I  /\  j  =  J ) )  -> 
( 0g `  R
)  =  ( 0g
`  R ) )
4038, 39, 6, 8, 2ovmpt2d 6425 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (
I ( 0g `  A ) J )  =  ( 0g `  R ) )
4140ad4antr 731 . . . . . . . 8  |-  ( ( ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  s  <  x )  /\  (
(coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  ( I
( 0g `  A
) J )  =  ( 0g `  R
) )
4232, 34, 413eqtrd 2512 . . . . . . 7  |-  ( ( ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  /\  s  <  x )  /\  (
(coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  [_ x  / 
k ]_ ( I ( (coe1 `  O ) `  k ) J )  =  ( 0g `  R ) )
4342exp31 604 . . . . . 6  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
s  <  x  ->  ( ( (coe1 `  O ) `  x )  =  ( 0g `  A )  ->  [_ x  /  k ]_ ( I ( (coe1 `  O ) `  k
) J )  =  ( 0g `  R
) ) ) )
4443a2d 26 . . . . 5  |-  ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( s  <  x  ->  ( (coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  ( s  <  x  ->  [_ x  / 
k ]_ ( I ( (coe1 `  O ) `  k ) J )  =  ( 0g `  R ) ) ) )
4544ralimdva 2875 . . . 4  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  s  e.  NN0 )  ->  ( A. x  e.  NN0  ( s  <  x  ->  (
(coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  A. x  e.  NN0  ( s  < 
x  ->  [_ x  / 
k ]_ ( I ( (coe1 `  O ) `  k ) J )  =  ( 0g `  R ) ) ) )
4645reximdva 2942 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  ( E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  ( (coe1 `  O ) `  x )  =  ( 0g `  A ) )  ->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  [_ x  / 
k ]_ ( I ( (coe1 `  O ) `  k ) J )  =  ( 0g `  R ) ) ) )
4727, 46mpd 15 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  [_ x  / 
k ]_ ( I ( (coe1 `  O ) `  k ) J )  =  ( 0g `  R ) ) )
482, 17, 47mptnn0fsupp 12083 1  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (
k  e.  NN0  |->  ( I ( (coe1 `  O ) `  k ) J ) ) finSupp  ( 0g `  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118   [_csb 3440   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297    ^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-fal 1385  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-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  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:  mply1topmatcllem  19173  mp2pm2mplem2  19177
  Copyright terms: Public domain W3C validator