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Theorem cpmidgsum2 19562
Description: Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
Hypotheses
Ref Expression
cpmadugsum.a  |-  A  =  ( N Mat  R )
cpmadugsum.b  |-  B  =  ( Base `  A
)
cpmadugsum.p  |-  P  =  (Poly1 `  R )
cpmadugsum.y  |-  Y  =  ( N Mat  P )
cpmadugsum.t  |-  T  =  ( N matToPolyMat  R )
cpmadugsum.x  |-  X  =  (var1 `  R )
cpmadugsum.e  |-  .^  =  (.g
`  (mulGrp `  P )
)
cpmadugsum.m  |-  .x.  =  ( .s `  Y )
cpmadugsum.r  |-  .X.  =  ( .r `  Y )
cpmadugsum.1  |-  .1.  =  ( 1r `  Y )
cpmadugsum.g  |-  .+  =  ( +g  `  Y )
cpmadugsum.s  |-  .-  =  ( -g `  Y )
cpmadugsum.i  |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `
 M ) )
cpmadugsum.j  |-  J  =  ( N maAdju  P )
cpmadugsum.0  |-  .0.  =  ( 0g `  Y )
cpmadugsum.g2  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `  0 ) ) ) ) ,  if ( n  =  (
s  +  1 ) ,  ( T `  ( b `  s
) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `  ( b `
 ( n  - 
1 ) ) ) 
.-  ( ( T `
 M )  .X.  ( T `  ( b `
 n ) ) ) ) ) ) ) )
cpmidgsum2.c  |-  C  =  ( N CharPlyMat  R )
cpmidgsum2.k  |-  K  =  ( C `  M
)
cpmidgsum2.h  |-  H  =  ( K  .x.  .1.  )
Assertion
Ref Expression
cpmidgsum2  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
0 ... s ) ) H  =  ( Y 
gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )
Distinct variable groups:    B, i    i, M    i, N    R, i    i, X    i, Y    .X. , i    .x. , i    .1. , i    i, b, s, T    .^ , i    .- , i    A, b, n, s    B, b, n, s    I,
b, i, n, s    J, b, i, n, s    M, b, n, s    N, b, n, s    P, i, n    R, b, n, s    T, b, n, s    X, b, n, s    Y, b, n, s    .^ , n, s, b    .x. , b, n, s    i, G    .X. , n    .0. ,
n    .- , n
Allowed substitution hints:    A( i)    C( i, n, s, b)    P( s, b)    .+ ( i, n, s, b)    .X. ( s, b)    .1. ( n, s, b)    G( n, s, b)    H( i, n, s, b)    K( i, n, s, b)    .- ( s,
b)    .0. ( i, s, b)

Proof of Theorem cpmidgsum2
StepHypRef Expression
1 cpmadugsum.a . . 3  |-  A  =  ( N Mat  R )
2 cpmadugsum.b . . 3  |-  B  =  ( Base `  A
)
3 cpmadugsum.p . . 3  |-  P  =  (Poly1 `  R )
4 cpmadugsum.y . . 3  |-  Y  =  ( N Mat  P )
5 cpmadugsum.t . . 3  |-  T  =  ( N matToPolyMat  R )
6 cpmadugsum.x . . 3  |-  X  =  (var1 `  R )
7 cpmadugsum.e . . 3  |-  .^  =  (.g
`  (mulGrp `  P )
)
8 cpmadugsum.m . . 3  |-  .x.  =  ( .s `  Y )
9 cpmadugsum.r . . 3  |-  .X.  =  ( .r `  Y )
10 cpmadugsum.1 . . 3  |-  .1.  =  ( 1r `  Y )
11 cpmadugsum.g . . 3  |-  .+  =  ( +g  `  Y )
12 cpmadugsum.s . . 3  |-  .-  =  ( -g `  Y )
13 cpmadugsum.i . . 3  |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `
 M ) )
14 cpmadugsum.j . . 3  |-  J  =  ( N maAdju  P )
15 cpmadugsum.0 . . 3  |-  .0.  =  ( 0g `  Y )
16 cpmadugsum.g2 . . 3  |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `  0 ) ) ) ) ,  if ( n  =  (
s  +  1 ) ,  ( T `  ( b `  s
) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `  ( b `
 ( n  - 
1 ) ) ) 
.-  ( ( T `
 M )  .X.  ( T `  ( b `
 n ) ) ) ) ) ) ) )
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16cpmadugsum 19561 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
0 ... s ) ) ( I  .X.  ( J `  I )
)  =  ( Y 
gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )
18 crngring 17419 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  R  e.  Ring )
1918anim2i 567 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
20193adant3 1015 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
213, 4pmatring 19376 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  Y  e.  Ring )
22 ringgrp 17413 . . . . . . . . . . 11  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
2320, 21, 223syl 20 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  Y  e.  Grp )
243, 4pmatlmod 19377 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  Y  e.  LMod )
2518, 24sylan2 472 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  Y  e.  LMod )
2618adantl 464 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  e.  Ring )
27 eqid 2400 . . . . . . . . . . . . . . 15  |-  ( Base `  P )  =  (
Base `  P )
286, 3, 27vr1cl 18468 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  X  e.  ( Base `  P
) )
2926, 28syl 17 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  X  e.  ( Base `  P ) )
303ply1crng 18447 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  P  e.  CRing
)
314matsca2 19104 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  Fin  /\  P  e.  CRing )  ->  P  =  (Scalar `  Y
) )
3230, 31sylan2 472 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  P  =  (Scalar `  Y
) )
3332fveq2d 5807 . . . . . . . . . . . . 13  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( Base `  P )  =  ( Base `  (Scalar `  Y ) ) )
3429, 33eleqtrd 2490 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  X  e.  ( Base `  (Scalar `  Y )
) )
35 eqid 2400 . . . . . . . . . . . . . 14  |-  ( Base `  Y )  =  (
Base `  Y )
3635, 10ringidcl 17429 . . . . . . . . . . . . 13  |-  ( Y  e.  Ring  ->  .1.  e.  ( Base `  Y )
)
3719, 21, 363syl 20 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  .1.  e.  ( Base `  Y
) )
38 eqid 2400 . . . . . . . . . . . . 13  |-  (Scalar `  Y )  =  (Scalar `  Y )
39 eqid 2400 . . . . . . . . . . . . 13  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
4035, 38, 8, 39lmodvscl 17739 . . . . . . . . . . . 12  |-  ( ( Y  e.  LMod  /\  X  e.  ( Base `  (Scalar `  Y ) )  /\  .1.  e.  ( Base `  Y
) )  ->  ( X  .x.  .1.  )  e.  ( Base `  Y
) )
4125, 34, 37, 40syl3anc 1228 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( X  .x.  .1.  )  e.  ( Base `  Y ) )
42413adant3 1015 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( X  .x.  .1.  )  e.  ( Base `  Y
) )
435, 1, 2, 3, 4mat2pmatbas 19409 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  ( T `  M )  e.  ( Base `  Y
) )
4418, 43syl3an2 1262 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( T `  M )  e.  ( Base `  Y
) )
4535, 12grpsubcl 16332 . . . . . . . . . 10  |-  ( ( Y  e.  Grp  /\  ( X  .x.  .1.  )  e.  ( Base `  Y
)  /\  ( T `  M )  e.  (
Base `  Y )
)  ->  ( ( X  .x.  .1.  )  .-  ( T `  M ) )  e.  ( Base `  Y ) )
4623, 42, 44, 45syl3anc 1228 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
( X  .x.  .1.  )  .-  ( T `  M ) )  e.  ( Base `  Y
) )
47303ad2ant2 1017 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  P  e.  CRing )
48 eqid 2400 . . . . . . . . . 10  |-  ( N maDet 
P )  =  ( N maDet  P )
494, 35, 14, 48, 10, 9, 8madurid 19328 . . . . . . . . 9  |-  ( ( ( ( X  .x.  .1.  )  .-  ( T `
 M ) )  e.  ( Base `  Y
)  /\  P  e.  CRing
)  ->  ( (
( X  .x.  .1.  )  .-  ( T `  M ) )  .X.  ( J `  ( ( X  .x.  .1.  )  .-  ( T `  M
) ) ) )  =  ( ( ( N maDet  P ) `  ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) )  .x.  .1.  )
)
5046, 47, 49syl2anc 659 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
( ( X  .x.  .1.  )  .-  ( T `
 M ) ) 
.X.  ( J `  ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) ) )  =  ( ( ( N maDet  P
) `  ( ( X  .x.  .1.  )  .-  ( T `  M ) ) )  .x.  .1.  ) )
51 id 22 . . . . . . . . . 10  |-  ( I  =  ( ( X 
.x.  .1.  )  .-  ( T `  M ) )  ->  I  =  ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) )
52 fveq2 5803 . . . . . . . . . 10  |-  ( I  =  ( ( X 
.x.  .1.  )  .-  ( T `  M ) )  ->  ( J `  I )  =  ( J `  ( ( X  .x.  .1.  )  .-  ( T `  M
) ) ) )
5351, 52oveq12d 6250 . . . . . . . . 9  |-  ( I  =  ( ( X 
.x.  .1.  )  .-  ( T `  M ) )  ->  ( I  .X.  ( J `  I
) )  =  ( ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) 
.X.  ( J `  ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) ) ) )
5413, 53mp1i 13 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
I  .X.  ( J `  I ) )  =  ( ( ( X 
.x.  .1.  )  .-  ( T `  M ) )  .X.  ( J `  ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) ) ) )
55 cpmidgsum2.h . . . . . . . . 9  |-  H  =  ( K  .x.  .1.  )
56 cpmidgsum2.k . . . . . . . . . . 11  |-  K  =  ( C `  M
)
57 cpmidgsum2.c . . . . . . . . . . . 12  |-  C  =  ( N CharPlyMat  R )
5857, 1, 2, 3, 4, 48, 12, 6, 8, 5, 10chpmatval 19514 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( C `  M )  =  ( ( N maDet 
P ) `  (
( X  .x.  .1.  )  .-  ( T `  M ) ) ) )
5956, 58syl5eq 2453 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  K  =  ( ( N maDet 
P ) `  (
( X  .x.  .1.  )  .-  ( T `  M ) ) ) )
6059oveq1d 6247 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( K  .x.  .1.  )  =  ( ( ( N maDet 
P ) `  (
( X  .x.  .1.  )  .-  ( T `  M ) ) ) 
.x.  .1.  ) )
6155, 60syl5eq 2453 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  H  =  ( ( ( N maDet  P ) `  ( ( X  .x.  .1.  )  .-  ( T `
 M ) ) )  .x.  .1.  )
)
6250, 54, 613eqtr4rd 2452 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  H  =  ( I  .X.  ( J `  I ) ) )
6362adantr 463 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  .X.  ( J `  I ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )  ->  H  =  ( I  .X.  ( J `  I ) ) )
64 simpr 459 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  .X.  ( J `  I ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )  ->  ( I  .X.  ( J `  I ) )  =  ( Y 
gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )
6563, 64eqtrd 2441 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
I  .X.  ( J `  I ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )  ->  H  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )
6665ex 432 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  (
( I  .X.  ( J `  I )
)  =  ( Y 
gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) )  ->  H  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
) ) ) ) ) )
6766reximdv 2875 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( E. b  e.  ( B  ^m  ( 0 ... s ) ) ( I  .X.  ( J `  I ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) )  ->  E. b  e.  ( B  ^m  ( 0 ... s ) ) H  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
) ) ) ) ) )
6867reximdv 2875 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( E. s  e.  NN  E. b  e.  ( B  ^m  ( 0 ... s ) ) ( I  .X.  ( J `  I ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  ( 0 ... s ) ) H  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
) ) ) ) ) )
6917, 68mpd 15 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
0 ... s ) ) H  =  ( Y 
gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   E.wrex 2752   ifcif 3882   class class class wbr 4392    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232    ^m cmap 7375   Fincfn 7472   0cc0 9440   1c1 9441    + caddc 9443    < clt 9576    - cmin 9759   NNcn 10494   NN0cn0 10754   ...cfz 11641   Basecbs 14731   +g cplusg 14799   .rcmulr 14800  Scalarcsca 14802   .scvsca 14803   0gc0g 14944    gsumg cgsu 14945   Grpcgrp 16267   -gcsg 16269  .gcmg 16270  mulGrpcmgp 17351   1rcur 17363   Ringcrg 17408   CRingccrg 17409   LModclmod 17722  var1cv1 18425  Poly1cpl1 18426   Mat cmat 19091   maDet cmdat 19268   maAdju cmadu 19316   matToPolyMat cmat2pmat 19387   CharPlyMat cchpmat 19509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-xor 1365  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-ot 3978  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-ofr 6476  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-tpos 6910  df-cur 6951  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-rp 11182  df-fz 11642  df-fzo 11766  df-seq 12060  df-exp 12119  df-hash 12358  df-word 12496  df-lsw 12497  df-concat 12498  df-s1 12499  df-substr 12500  df-splice 12501  df-reverse 12502  df-s2 12774  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-0g 14946  df-gsum 14947  df-prds 14952  df-pws 14954  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-mhm 16180  df-submnd 16181  df-grp 16271  df-minusg 16272  df-sbg 16273  df-mulg 16274  df-subg 16412  df-ghm 16479  df-gim 16521  df-cntz 16569  df-oppg 16595  df-symg 16617  df-pmtr 16681  df-psgn 16730  df-evpm 16731  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-srg 17368  df-ring 17410  df-cring 17411  df-oppr 17482  df-dvdsr 17500  df-unit 17501  df-invr 17531  df-dvr 17542  df-rnghom 17574  df-drng 17608  df-subrg 17637  df-lmod 17724  df-lss 17789  df-sra 18028  df-rgmod 18029  df-assa 18171  df-ascl 18173  df-psr 18215  df-mvr 18216  df-mpl 18217  df-opsr 18219  df-psr1 18429  df-vr1 18430  df-ply1 18431  df-coe1 18432  df-cnfld 18631  df-zring 18699  df-zrh 18731  df-dsmm 18951  df-frlm 18966  df-mamu 19068  df-mat 19092  df-mdet 19269  df-madu 19318  df-mat2pmat 19390  df-decpmat 19446  df-chpmat 19510
This theorem is referenced by:  cpmidg2sum  19563
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