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Theorem mptcoe1matfsupp 19757
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 5891 . . 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 2429 . . 3  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2429 . . 3  |-  ( Base `  A )  =  (
Base `  A )
6 simp2 1006 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  I  e.  N )
76adantr 466 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  I  e.  N )
8 simp3 1007 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  J  e.  N )
98adantr 466 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring  /\  O  e.  L
)  /\  I  e.  N  /\  J  e.  N
)  /\  k  e.  NN0 )  ->  J  e.  N )
10 simp3 1007 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  O  e.  L )
11103ad2ant1 1026 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  O  e.  L )
12 eqid 2429 . . . . 5  |-  (coe1 `  O
)  =  (coe1 `  O
)
13 mptcoe1matfsupp.l . . . . 5  |-  L  =  ( Base `  Q
)
14 mptcoe1matfsupp.q . . . . 5  |-  Q  =  (Poly1 `  A )
1512, 13, 14, 5coe1fvalcl 18740 . . . 4  |-  ( ( O  e.  L  /\  k  e.  NN0 )  -> 
( (coe1 `  O ) `  k )  e.  (
Base `  A )
)
1611, 15sylan 473 . . 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 19382 . 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 2429 . . . . . . 7  |-  ( 0g
`  A )  =  ( 0g `  A
)
1912, 13, 14, 18, 5coe1fsupp 18742 . . . . . 6  |-  ( O  e.  L  ->  (coe1 `  O )  e.  {
c  e.  ( (
Base `  A )  ^m  NN0 )  |  c finSupp 
( 0g `  A
) } )
20 elrabi 3232 . . . . . 6  |-  ( (coe1 `  O )  e.  {
c  e.  ( (
Base `  A )  ^m  NN0 )  |  c finSupp 
( 0g `  A
) }  ->  (coe1 `  O )  e.  ( ( Base `  A
)  ^m  NN0 ) )
2111, 19, 203syl 18 . . . . 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 5891 . . . . 5  |-  ( 0g
`  A )  e. 
_V
2321, 22jctir 540 . . . 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 18741 . . . . 5  |-  ( O  e.  L  ->  (coe1 `  O ) finSupp  ( 0g `  A ) )
2511, 24syl 17 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (coe1 `  O ) finSupp  ( 0g `  A ) )
26 fsuppmapnn0ub 12204 . . . 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 62 . . 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 6340 . . . . . . . . . 10  |-  [_ x  /  k ]_ (
I ( (coe1 `  O
) `  k ) J )  =  ( I [_ x  / 
k ]_ ( (coe1 `  O
) `  k ) J )
29 csbfv 5918 . . . . . . . . . . 11  |-  [_ x  /  k ]_ (
(coe1 `  O ) `  k )  =  ( (coe1 `  O ) `  x )
3029oveqi 6318 . . . . . . . . . 10  |-  ( I
[_ x  /  k ]_ ( (coe1 `  O ) `  k ) J )  =  ( I ( (coe1 `  O ) `  x ) J )
3128, 30eqtri 2458 . . . . . . . . 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 6311 . . . . . . . . 9  |-  ( ( (coe1 `  O ) `  x )  =  ( 0g `  A )  ->  ( I ( (coe1 `  O ) `  x ) J )  =  ( I ( 0g `  A ) J ) )
3433adantl 467 . . . . . . . 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 2429 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
363, 35mat0op 19375 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  A
)  =  ( i  e.  N ,  j  e.  N  |->  ( 0g
`  R ) ) )
37363adant3 1025 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  ( 0g `  A )  =  ( i  e.  N ,  j  e.  N  |->  ( 0g `  R
) ) )
38373ad2ant1 1026 . . . . . . . . . 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 2430 . . . . . . . . . 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 6438 . . . . . . . . 9  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  I  e.  N  /\  J  e.  N )  ->  (
I ( 0g `  A ) J )  =  ( 0g `  R ) )
4140ad4antr 736 . . . . . . . 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 2474 . . . . . . 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 607 . . . . . 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 29 . . . . 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 2840 . . . 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 2907 . . 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 12206 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   {crab 2786   _Vcvv 3087   [_csb 3401   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307    ^m cmap 7480   Fincfn 7577   finSupp cfsupp 7889    < clt 9674   NN0cn0 10869   Basecbs 15084   0gc0g 15297   Ringcrg 17715  Poly1cpl1 18705  coe1cco1 18706   Mat cmat 19363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-0g 15299  df-prds 15305  df-pws 15307  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-mgp 17659  df-ur 17671  df-ring 17717  df-subrg 17941  df-lmod 18028  df-lss 18091  df-sra 18330  df-rgmod 18331  df-psr 18515  df-mpl 18517  df-opsr 18519  df-psr1 18708  df-ply1 18710  df-coe1 18711  df-dsmm 19226  df-frlm 19241  df-mat 19364
This theorem is referenced by:  mply1topmatcllem  19758
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