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Theorem mptcoe1matfsupp 31252
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.)
Hypotheses
Ref Expression
mptcoe1matfsupp.a  |-  A  =  ( N Mat  R )
mptcoe1matfsupp.q  |-  Q  =  (Poly1 `  A )
mptcoe1matfsupp.b  |-  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 eqid 2451 . 2  |-  ( 0g
`  R )  =  ( 0g `  R
)
2 simp2 989 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  I  e.  N )
32adantr 465 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  I  e.  N )
4 simp3 990 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  J  e.  N )
54adantr 465 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  J  e.  N )
6 simp3 990 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  O  e.  L )
763ad2ant1 1009 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  O  e.  L )
8 eqid 2451 . . . . 5  |-  (coe1 `  O
)  =  (coe1 `  O
)
9 mptcoe1matfsupp.b . . . . 5  |-  L  =  ( Base `  Q
)
10 mptcoe1matfsupp.q . . . . 5  |-  Q  =  (Poly1 `  A )
11 eqid 2451 . . . . 5  |-  ( Base `  A )  =  (
Base `  A )
128, 9, 10, 11coe1fvalcl 30975 . . . 4  |-  ( ( O  e.  L  /\  k  e.  NN0 )  -> 
( (coe1 `  O ) `  k )  e.  (
Base `  A )
)
137, 12sylan 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 ) )
14 mptcoe1matfsupp.a . . . 4  |-  A  =  ( N Mat  R )
15 eqid 2451 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
1614, 15matecl 18444 . . 3  |-  ( ( I  e.  N  /\  J  e.  N  /\  ( (coe1 `  O ) `  k )  e.  (
Base `  A )
)  ->  ( I
( (coe1 `  O ) `  k ) J )  e.  ( Base `  R
) )
173, 5, 13, 16syl3anc 1219 . 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 2451 . . . . . . 7  |-  ( 0g
`  A )  =  ( 0g `  A
)
198, 9, 10, 11, 18coe1fsupp 30976 . . . . . 6  |-  ( O  e.  L  ->  (coe1 `  O )  e.  {
c  e.  ( (
Base `  A )  ^m  NN0 )  |  c finSupp 
( 0g `  A
) } )
20 elrabi 3214 . . . . . 6  |-  ( (coe1 `  O )  e.  {
c  e.  ( (
Base `  A )  ^m  NN0 )  |  c finSupp 
( 0g `  A
) }  ->  (coe1 `  O )  e.  ( ( Base `  A
)  ^m  NN0 ) )
217, 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 5802 . . . . 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 ) )
248, 9, 10, 18coe1sfi 17784 . . . . 5  |-  ( O  e.  L  ->  (coe1 `  O ) finSupp  ( 0g `  A ) )
257, 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 30937 . . . 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 6225 . . . . . . . . . 10  |-  [_ x  /  k ]_ (
I ( (coe1 `  O
) `  k ) J )  =  ( I [_ x  / 
k ]_ ( (coe1 `  O
) `  k ) J )
29 csbfv 5830 . . . . . . . . . . 11  |-  [_ x  /  k ]_ (
(coe1 `  O ) `  k )  =  ( (coe1 `  O ) `  x )
3029oveqi 6206 . . . . . . . . . 10  |-  ( I
[_ x  /  k ]_ ( (coe1 `  O ) `  k ) J )  =  ( I ( (coe1 `  O ) `  x ) J )
3128, 30eqtri 2480 . . . . . . . . 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 6199 . . . . . . . . 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 ) )
3514, 1mat0op 18438 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( i  e.  N ,  j  e.  N  |->  ( 0g
`  R ) ) )
36353adant3 1008 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( 0g `  A )  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R
) ) )
37363ad2ant1 1009 . . . . . . . . . 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
) ) )
38 eqidd 2452 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  ( i  =  I  /\  j  =  J ) )  -> 
( 0g `  R
)  =  ( 0g
`  R ) )
39 fvex 5802 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
4039a1i 11 . . . . . . . . . 10  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  ( 0g `  R )  e. 
_V )
4137, 38, 2, 4, 40ovmpt2d 6321 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (
I ( 0g `  A ) J )  =  ( 0g `  R ) )
4241ad4antr 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
) )
4332, 34, 423eqtrd 2496 . . . . . . 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 ) )
4443exp31 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
) ) ) )
4544a2d 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 ) ) ) )
4645ralimdva 2827 . . . 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 ) ) ) )
4746reximdva 2927 . . 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 ) ) ) )
4827, 47mpd 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 ) ) )
491, 17, 48mptnn0fsupp 30943 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3071   [_csb 3389   class class class wbr 4393    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195    ^m cmap 7317   Fincfn 7413   finSupp cfsupp 7724    < clt 9522   NN0cn0 10683   Basecbs 14285   0gc0g 14489   Ringcrg 16760  Poly1cpl1 17749  coe1cco1 17750   Mat cmat 18398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-hom 14373  df-cco 14374  df-0g 14491  df-prds 14497  df-pws 14499  df-mnd 15526  df-grp 15656  df-minusg 15657  df-sbg 15658  df-subg 15789  df-mgp 16706  df-ur 16718  df-rng 16762  df-subrg 16978  df-lmod 17065  df-lss 17129  df-sra 17368  df-rgmod 17369  df-psr 17538  df-mpl 17540  df-opsr 17542  df-psr1 17752  df-ply1 17754  df-coe1 17755  df-dsmm 18275  df-frlm 18290  df-mat 18400
This theorem is referenced by:  mply1topmatcllem  31253  mp2pm2mplem2  31265
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