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Theorem cpmidpmatlem3 19133
Description: Lemma 3 for cpmidpmat 19134. (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 5867 . . . 4  |-  ( 0g
`  A )  e. 
_V
32a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( 0g `  A )  e. 
_V )
4 ovex 6300 . . . 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 5857 . . . 4  |-  ( k  =  l  ->  (
(coe1 `  K ) `  k )  =  ( (coe1 `  K ) `  l ) )
76oveq1d 6290 . . 3  |-  ( k  =  l  ->  (
( (coe1 `  K ) `  k )  .*  O
)  =  ( ( (coe1 `  K ) `  l )  .*  O
) )
8 fvex 5867 . . . . . 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 2460 . . . . . . . 8  |-  ( Base `  P )  =  (
Base `  P )
1611, 12, 13, 14, 15chpmatply1 19093 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( C `  M )  e.  ( Base `  P
) )
1710, 16syl5eqel 2552 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  K  e.  ( Base `  P
) )
18 eqid 2460 . . . . . . 7  |-  (coe1 `  K
)  =  (coe1 `  K
)
19 eqid 2460 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
2018, 15, 14, 19coe1fvalcl 18015 . . . . . 6  |-  ( ( K  e.  ( Base `  P )  /\  n  e.  NN0 )  ->  (
(coe1 `  K ) `  n )  e.  (
Base `  R )
)
2117, 20sylan 471 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  n  e.  NN0 )  ->  (
(coe1 `  K ) `  n )  e.  (
Base `  R )
)
22 crngrng 16989 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
23223ad2ant2 1013 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  R  e.  Ring )
24 eqid 2460 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
2514, 15, 24mptcoe1fsupp 18019 . . . . . 6  |-  ( ( R  e.  Ring  /\  K  e.  ( Base `  P
) )  ->  (
n  e.  NN0  |->  ( (coe1 `  K ) `  n
) ) finSupp  ( 0g `  R ) )
2623, 17, 25syl2anc 661 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
n  e.  NN0  |->  ( (coe1 `  K ) `  n
) ) finSupp  ( 0g `  R ) )
279, 21, 26mptnn0fsuppr 12061 . . . 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 5895 . . . . . . . . . . . . . 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 2462 . . . . . . . . . . . 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 484 . . . . . . . . . . 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 18682 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
33323adant3 1011 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  R  =  (Scalar `  A )
)
3433ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  CRing  /\  M  e.  B
)  /\  l  e.  NN0 )  /\  [_ l  /  n ]_ ( (coe1 `  K ) `  n
)  =  ( 0g
`  R ) )  ->  R  =  (Scalar `  A ) )
3534fveq2d 5861 . . . . . . . . . . 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 2501 . . . . . . . . . 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 6290 . . . . . . . . 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 18691 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
3922, 38sylan2 474 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
40393adant3 1011 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  A  e.  LMod )
4112matrng 18705 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
4222, 41sylan2 474 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
43 cpmidgsumm2pm.o . . . . . . . . . . . . . 14  |-  O  =  ( 1r `  A
)
4413, 43rngidcl 16999 . . . . . . . . . . . . 13  |-  ( A  e.  Ring  ->  O  e.  B )
4542, 44syl 16 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  O  e.  B )
46453adant3 1011 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  O  e.  B )
47 eqid 2460 . . . . . . . . . . . 12  |-  (Scalar `  A )  =  (Scalar `  A )
48 cpmidgsumm2pm.m . . . . . . . . . . . 12  |-  .*  =  ( .s `  A )
49 eqid 2460 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  A )
)  =  ( 0g
`  (Scalar `  A )
)
50 eqid 2460 . . . . . . . . . . . 12  |-  ( 0g
`  A )  =  ( 0g `  A
)
5113, 47, 48, 49, 50lmod0vs 17321 . . . . . . . . . . 11  |-  ( ( A  e.  LMod  /\  O  e.  B )  ->  (
( 0g `  (Scalar `  A ) )  .*  O )  =  ( 0g `  A ) )
5240, 46, 51syl2anc 661 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
( 0g `  (Scalar `  A ) )  .*  O )  =  ( 0g `  A ) )
5352ad2antrr 725 . . . . . . . . 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 2501 . . . . . . . 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 434 . . . . . . 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 52 . . . . . 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 2865 . . . . 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 2930 . . . 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 12060 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
k  e.  NN0  |->  ( ( (coe1 `  K ) `  k )  .*  O
) ) finSupp  ( 0g `  A ) )
611, 60syl5eqbr 4473 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   _Vcvv 3106   [_csb 3428   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Fincfn 7506   finSupp cfsupp 7818    < clt 9617   NN0cn0 10784   Basecbs 14479  Scalarcsca 14547   .scvsca 14548   0gc0g 14684  .gcmg 15720  mulGrpcmgp 16924   1rcur 16936   Ringcrg 16979   CRingccrg 16980   LModclmod 17288  algSccascl 17724  var1cv1 17979  Poly1cpl1 17980  coe1cco1 17981   Mat cmat 18669   matToPolyMat cmat2pmat 18965   CharPlyMat cchpmat 19087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-ofr 6516  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-substr 12499  df-splice 12500  df-reverse 12501  df-s2 12763  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-0g 14686  df-gsum 14687  df-prds 14692  df-pws 14694  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-ghm 16053  df-gim 16095  df-cntz 16143  df-oppg 16169  df-symg 16191  df-pmtr 16256  df-psgn 16305  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-rnghom 17141  df-drng 17174  df-subrg 17203  df-lmod 17290  df-lss 17355  df-sra 17594  df-rgmod 17595  df-ascl 17727  df-psr 17769  df-mvr 17770  df-mpl 17771  df-opsr 17773  df-psr1 17983  df-vr1 17984  df-ply1 17985  df-coe1 17986  df-cnfld 18185  df-zring 18250  df-zrh 18301  df-dsmm 18523  df-frlm 18538  df-mamu 18646  df-mat 18670  df-mdet 18847  df-mat2pmat 18968  df-chpmat 19088
This theorem is referenced by:  cpmidpmat  19134
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