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Theorem cpmidpmatlem3 19665
Description: Lemma 3 for cpmidpmat 19666. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
Hypotheses
Ref Expression
cpmidgsum.a  |-  A  =  ( N Mat  R )
cpmidgsum.b  |-  B  =  ( Base `  A
)
cpmidgsum.p  |-  P  =  (Poly1 `  R )
cpmidgsum.y  |-  Y  =  ( N Mat  P )
cpmidgsum.x  |-  X  =  (var1 `  R )
cpmidgsum.e  |-  .^  =  (.g
`  (mulGrp `  P )
)
cpmidgsum.m  |-  .x.  =  ( .s `  Y )
cpmidgsum.1  |-  .1.  =  ( 1r `  Y )
cpmidgsum.u  |-  U  =  (algSc `  P )
cpmidgsum.c  |-  C  =  ( N CharPlyMat  R )
cpmidgsum.k  |-  K  =  ( C `  M
)
cpmidgsum.h  |-  H  =  ( K  .x.  .1.  )
cpmidgsumm2pm.o  |-  O  =  ( 1r `  A
)
cpmidgsumm2pm.m  |-  .*  =  ( .s `  A )
cpmidgsumm2pm.t  |-  T  =  ( N matToPolyMat  R )
cpmidpmat.g  |-  G  =  ( k  e.  NN0  |->  ( ( (coe1 `  K
) `  k )  .*  O ) )
Assertion
Ref Expression
cpmidpmatlem3  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  G finSupp  ( 0g `  A ) )
Distinct variable groups:    B, k    k, H    k, N    P, k    R, k    k, Y   
k, K    k, O    .* , k    k, M
Allowed substitution hints:    A( k)    C( k)    T( k)    .x. ( k)    U( k)    .1. ( k)    .^ ( k)    G( k)    X( k)

Proof of Theorem cpmidpmatlem3
Dummy variables  n  l  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cpmidpmat.g . 2  |-  G  =  ( k  e.  NN0  |->  ( ( (coe1 `  K
) `  k )  .*  O ) )
2 fvex 5859 . . . 4  |-  ( 0g
`  A )  e. 
_V
32a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( 0g `  A )  e. 
_V )
4 ovex 6306 . . . 4  |-  ( ( (coe1 `  K ) `  k )  .*  O
)  e.  _V
54a1i 11 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  k  e.  NN0 )  ->  (
( (coe1 `  K ) `  k )  .*  O
)  e.  _V )
6 fveq2 5849 . . . 4  |-  ( k  =  l  ->  (
(coe1 `  K ) `  k )  =  ( (coe1 `  K ) `  l ) )
76oveq1d 6293 . . 3  |-  ( k  =  l  ->  (
( (coe1 `  K ) `  k )  .*  O
)  =  ( ( (coe1 `  K ) `  l )  .*  O
) )
8 fvex 5859 . . . . . 6  |-  ( 0g
`  R )  e. 
_V
98a1i 11 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( 0g `  R )  e. 
_V )
10 cpmidgsum.k . . . . . . 7  |-  K  =  ( C `  M
)
11 cpmidgsum.c . . . . . . . 8  |-  C  =  ( N CharPlyMat  R )
12 cpmidgsum.a . . . . . . . 8  |-  A  =  ( N Mat  R )
13 cpmidgsum.b . . . . . . . 8  |-  B  =  ( Base `  A
)
14 cpmidgsum.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
15 eqid 2402 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
1611, 12, 13, 14, 15chpmatply1 19625 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( C `  M )  e.  ( Base `  P
) )
1710, 16syl5eqel 2494 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  K  e.  ( Base `  P
) )
18 eqid 2402 . . . . . . 7  |-  (coe1 `  K
)  =  (coe1 `  K
)
19 eqid 2402 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
2018, 15, 14, 19coe1fvalcl 18571 . . . . . 6  |-  ( ( K  e.  ( Base `  P )  /\  n  e.  NN0 )  ->  (
(coe1 `  K ) `  n )  e.  (
Base `  R )
)
2117, 20sylan 469 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  n  e.  NN0 )  ->  (
(coe1 `  K ) `  n )  e.  (
Base `  R )
)
22 crngring 17529 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
23223ad2ant2 1019 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  R  e.  Ring )
24 eqid 2402 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
2514, 15, 24mptcoe1fsupp 18575 . . . . . 6  |-  ( ( R  e.  Ring  /\  K  e.  ( Base `  P
) )  ->  (
n  e.  NN0  |->  ( (coe1 `  K ) `  n
) ) finSupp  ( 0g `  R ) )
2623, 17, 25syl2anc 659 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
n  e.  NN0  |->  ( (coe1 `  K ) `  n
) ) finSupp  ( 0g `  R ) )
279, 21, 26mptnn0fsuppr 12149 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN0  A. l  e. 
NN0  ( s  < 
l  ->  [_ l  /  n ]_ ( (coe1 `  K
) `  n )  =  ( 0g `  R ) ) )
28 csbfv 5886 . . . . . . . . . . . . . 14  |-  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( (coe1 `  K ) `  l
)
2928a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  l  e.  NN0 )  ->  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( (coe1 `  K ) `  l
) )
3029eqeq1d 2404 . . . . . . . . . . . 12  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  l  e.  NN0 )  ->  ( [_ l  /  n ]_ ( (coe1 `  K ) `  n )  =  ( 0g `  R )  <-> 
( (coe1 `  K ) `  l )  =  ( 0g `  R ) ) )
3130biimpa 482 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  ( (coe1 `  K
) `  l )  =  ( 0g `  R ) )
3212matsca2 19214 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
33323adant3 1017 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  R  =  (Scalar `  A )
)
3433ad2antrr 724 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  R  =  (Scalar `  A ) )
3534fveq2d 5853 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  ( 0g `  R )  =  ( 0g `  (Scalar `  A ) ) )
3631, 35eqtrd 2443 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  ( (coe1 `  K
) `  l )  =  ( 0g `  (Scalar `  A ) ) )
3736oveq1d 6293 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  ( ( (coe1 `  K ) `  l
)  .*  O )  =  ( ( 0g
`  (Scalar `  A )
)  .*  O ) )
3812matlmod 19223 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
3922, 38sylan2 472 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
40393adant3 1017 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  A  e.  LMod )
4112matring 19237 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
4222, 41sylan2 472 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
43 cpmidgsumm2pm.o . . . . . . . . . . . . . 14  |-  O  =  ( 1r `  A
)
4413, 43ringidcl 17539 . . . . . . . . . . . . 13  |-  ( A  e.  Ring  ->  O  e.  B )
4542, 44syl 17 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  O  e.  B )
46453adant3 1017 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  O  e.  B )
47 eqid 2402 . . . . . . . . . . . 12  |-  (Scalar `  A )  =  (Scalar `  A )
48 cpmidgsumm2pm.m . . . . . . . . . . . 12  |-  .*  =  ( .s `  A )
49 eqid 2402 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  A )
)  =  ( 0g
`  (Scalar `  A )
)
50 eqid 2402 . . . . . . . . . . . 12  |-  ( 0g
`  A )  =  ( 0g `  A
)
5113, 47, 48, 49, 50lmod0vs 17865 . . . . . . . . . . 11  |-  ( ( A  e.  LMod  /\  O  e.  B )  ->  (
( 0g `  (Scalar `  A ) )  .*  O )  =  ( 0g `  A ) )
5240, 46, 51syl2anc 659 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
( 0g `  (Scalar `  A ) )  .*  O )  =  ( 0g `  A ) )
5352ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  ( ( 0g
`  (Scalar `  A )
)  .*  O )  =  ( 0g `  A ) )
5437, 53eqtrd 2443 . . . . . . . 8  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  ( ( (coe1 `  K ) `  l
)  .*  O )  =  ( 0g `  A ) )
5554ex 432 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  l  e.  NN0 )  ->  ( [_ l  /  n ]_ ( (coe1 `  K ) `  n )  =  ( 0g `  R )  ->  ( ( (coe1 `  K ) `  l
)  .*  O )  =  ( 0g `  A ) ) )
5655imim2d 51 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  l  e.  NN0 )  ->  (
( s  <  l  ->  [_ l  /  n ]_ ( (coe1 `  K ) `  n )  =  ( 0g `  R ) )  ->  ( s  <  l  ->  ( (
(coe1 `  K ) `  l )  .*  O
)  =  ( 0g
`  A ) ) ) )
5756ralimdva 2812 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( A. l  e.  NN0  ( s  <  l  ->  [_ l  /  n ]_ ( (coe1 `  K ) `  n )  =  ( 0g `  R ) )  ->  A. l  e.  NN0  ( s  < 
l  ->  ( (
(coe1 `  K ) `  l )  .*  O
)  =  ( 0g
`  A ) ) ) )
5857reximdv 2878 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( E. s  e.  NN0  A. l  e.  NN0  (
s  <  l  ->  [_ l  /  n ]_ ( (coe1 `  K ) `  n )  =  ( 0g `  R ) )  ->  E. s  e.  NN0  A. l  e. 
NN0  ( s  < 
l  ->  ( (
(coe1 `  K ) `  l )  .*  O
)  =  ( 0g
`  A ) ) ) )
5927, 58mpd 15 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN0  A. l  e. 
NN0  ( s  < 
l  ->  ( (
(coe1 `  K ) `  l )  .*  O
)  =  ( 0g
`  A ) ) )
603, 5, 7, 59mptnn0fsuppd 12148 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
k  e.  NN0  |->  ( ( (coe1 `  K ) `  k )  .*  O
) ) finSupp  ( 0g `  A ) )
611, 60syl5eqbr 4428 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  G finSupp  ( 0g `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   _Vcvv 3059   [_csb 3373   class class class wbr 4395    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278   Fincfn 7554   finSupp cfsupp 7863    < clt 9658   NN0cn0 10836   Basecbs 14841  Scalarcsca 14912   .scvsca 14913   0gc0g 15054  .gcmg 16380  mulGrpcmgp 17461   1rcur 17473   Ringcrg 17518   CRingccrg 17519   LModclmod 17832  algSccascl 18280  var1cv1 18535  Poly1cpl1 18536  coe1cco1 18537   Mat cmat 19201   matToPolyMat cmat2pmat 19497   CharPlyMat cchpmat 19619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-xor 1367  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-ofr 6522  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-word 12591  df-lsw 12592  df-concat 12593  df-s1 12594  df-substr 12595  df-splice 12596  df-reverse 12597  df-s2 12869  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-0g 15056  df-gsum 15057  df-prds 15062  df-pws 15064  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-ghm 16589  df-gim 16631  df-cntz 16679  df-oppg 16705  df-symg 16727  df-pmtr 16791  df-psgn 16840  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-dvr 17652  df-rnghom 17684  df-drng 17718  df-subrg 17747  df-lmod 17834  df-lss 17899  df-sra 18138  df-rgmod 18139  df-ascl 18283  df-psr 18325  df-mvr 18326  df-mpl 18327  df-opsr 18329  df-psr1 18539  df-vr1 18540  df-ply1 18541  df-coe1 18542  df-cnfld 18741  df-zring 18809  df-zrh 18841  df-dsmm 19061  df-frlm 19076  df-mamu 19178  df-mat 19202  df-mdet 19379  df-mat2pmat 19500  df-chpmat 19620
This theorem is referenced by:  cpmidpmat  19666
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