MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mdetdiagid Structured version   Unicode version

Theorem mdetdiagid 18522
Description: The determinant of a diagonal matrix with identical entries is the power of the entry in the diagonal. (Contributed by AV, 17-Aug-2019.)
Hypotheses
Ref Expression
mdetdiag.d  |-  D  =  ( N maDet  R )
mdetdiag.a  |-  A  =  ( N Mat  R )
mdetdiag.b  |-  B  =  ( Base `  A
)
mdetdiag.g  |-  G  =  (mulGrp `  R )
mdetdiag.0  |-  .0.  =  ( 0g `  R )
mdetdiagid.c  |-  C  =  ( Base `  R
)
mdetdiagid.t  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
mdetdiagid  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( D `  M
)  =  ( (
# `  N )  .x.  X ) ) )
Distinct variable groups:    i, M, j    i, N, j    .0. , i, j    B, i, j    C, i, j    R, i, j    i, X, j
Allowed substitution hints:    A( i, j)    D( i, j)    .x. ( i, j)    G( i, j)

Proof of Theorem mdetdiagid
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  R  e.  CRing )
21adantr 465 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  R  e.  CRing )
3 simpr 461 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  N  e.  Fin )
43adantr 465 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  N  e.  Fin )
5 simpl 457 . . . . . . 7  |-  ( ( M  e.  B  /\  X  e.  C )  ->  M  e.  B )
65adantl 466 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  M  e.  B )
72, 4, 63jca 1168 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( R  e.  CRing  /\  N  e.  Fin  /\  M  e.  B ) )
87adantr 465 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( R  e.  CRing  /\  N  e.  Fin  /\  M  e.  B )
)
9 id 22 . . . . . . . . . 10  |-  ( ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )
10 ifnefalse 3899 . . . . . . . . . . 11  |-  ( i  =/=  j  ->  if ( i  =  j ,  X ,  .0.  )  =  .0.  )
1110adantl 466 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C )
)  /\  i  e.  N )  /\  j  e.  N )  /\  i  =/=  j )  ->  if ( i  =  j ,  X ,  .0.  )  =  .0.  )
129, 11sylan9eqr 2514 . . . . . . . . 9  |-  ( ( ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  /\  i  e.  N )  /\  j  e.  N )  /\  i  =/=  j )  /\  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( i M j )  =  .0.  )
1312exp31 604 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C )
)  /\  i  e.  N )  /\  j  e.  N )  ->  (
i  =/=  j  -> 
( ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( i M j )  =  .0.  ) ) )
1413com23 78 . . . . . . 7  |-  ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C )
)  /\  i  e.  N )  /\  j  e.  N )  ->  (
( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( i  =/=  j  ->  ( i M j )  =  .0.  )
) )
1514ralimdva 2824 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  i  e.  N )  ->  ( A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  A. j  e.  N  ( i  =/=  j  ->  ( i M j )  =  .0.  )
) )
1615ralimdva 2824 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i M j )  =  .0.  )
) )
1716imp 429 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  ->  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i M j )  =  .0.  )
)
18 mdetdiag.d . . . . 5  |-  D  =  ( N maDet  R )
19 mdetdiag.a . . . . 5  |-  A  =  ( N Mat  R )
20 mdetdiag.b . . . . 5  |-  B  =  ( Base `  A
)
21 mdetdiag.g . . . . 5  |-  G  =  (mulGrp `  R )
22 mdetdiag.0 . . . . 5  |-  .0.  =  ( 0g `  R )
2318, 19, 20, 21, 22mdetdiag 18521 . . . 4  |-  ( ( R  e.  CRing  /\  N  e.  Fin  /\  M  e.  B )  ->  ( A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i M j )  =  .0.  )  ->  ( D `  M
)  =  ( G 
gsumg  ( k  e.  N  |->  ( k M k ) ) ) ) )
248, 17, 23sylc 60 . . 3  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( D `  M
)  =  ( G 
gsumg  ( k  e.  N  |->  ( k M k ) ) ) )
25 oveq1 6197 . . . . . . . . . . . 12  |-  ( i  =  k  ->  (
i M j )  =  ( k M j ) )
26 equequ1 1738 . . . . . . . . . . . . 13  |-  ( i  =  k  ->  (
i  =  j  <->  k  =  j ) )
2726ifbid 3909 . . . . . . . . . . . 12  |-  ( i  =  k  ->  if ( i  =  j ,  X ,  .0.  )  =  if (
k  =  j ,  X ,  .0.  )
)
2825, 27eqeq12d 2473 . . . . . . . . . . 11  |-  ( i  =  k  ->  (
( i M j )  =  if ( i  =  j ,  X ,  .0.  )  <->  ( k M j )  =  if ( k  =  j ,  X ,  .0.  ) ) )
29 oveq2 6198 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
k M j )  =  ( k M k ) )
30 equequ2 1739 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
k  =  j  <->  k  =  k ) )
3130ifbid 3909 . . . . . . . . . . . 12  |-  ( j  =  k  ->  if ( k  =  j ,  X ,  .0.  )  =  if (
k  =  k ,  X ,  .0.  )
)
3229, 31eqeq12d 2473 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
( k M j )  =  if ( k  =  j ,  X ,  .0.  )  <->  ( k M k )  =  if ( k  =  k ,  X ,  .0.  ) ) )
3328, 32rspc2v 3176 . . . . . . . . . 10  |-  ( ( k  e.  N  /\  k  e.  N )  ->  ( A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( k M k )  =  if ( k  =  k ,  X ,  .0.  ) ) )
3433anidms 645 . . . . . . . . 9  |-  ( k  e.  N  ->  ( A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( k M k )  =  if ( k  =  k ,  X ,  .0.  )
) )
3534adantl 466 . . . . . . . 8  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  k  e.  N )  ->  ( A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( k M k )  =  if ( k  =  k ,  X ,  .0.  ) ) )
3635imp 429 . . . . . . 7  |-  ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C )
)  /\  k  e.  N )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( k M k )  =  if ( k  =  k ,  X ,  .0.  )
)
37 equid 1731 . . . . . . . 8  |-  k  =  k
3837iftruei 3896 . . . . . . 7  |-  if ( k  =  k ,  X ,  .0.  )  =  X
3936, 38syl6eq 2508 . . . . . 6  |-  ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C )
)  /\  k  e.  N )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( k M k )  =  X )
4039an32s 802 . . . . 5  |-  ( ( ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C )
)  /\  A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  /\  k  e.  N )  ->  (
k M k )  =  X )
4140mpteq2dva 4476 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( k  e.  N  |->  ( k M k ) )  =  ( k  e.  N  |->  X ) )
4241oveq2d 6206 . . 3  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( G  gsumg  ( k  e.  N  |->  ( k M k ) ) )  =  ( G  gsumg  ( k  e.  N  |->  X ) ) )
4321crngmgp 16759 . . . . . . . 8  |-  ( R  e.  CRing  ->  G  e. CMnd )
44 cmnmnd 16396 . . . . . . . 8  |-  ( G  e. CMnd  ->  G  e.  Mnd )
4543, 44syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  G  e.  Mnd )
4645adantr 465 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  G  e.  Mnd )
4746adantr 465 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  G  e.  Mnd )
48 simpr 461 . . . . . 6  |-  ( ( M  e.  B  /\  X  e.  C )  ->  X  e.  C )
4948adantl 466 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  X  e.  C )
50 mdetdiagid.c . . . . . . 7  |-  C  =  ( Base `  R
)
5121, 50mgpbas 16702 . . . . . 6  |-  C  =  ( Base `  G
)
52 mdetdiagid.t . . . . . 6  |-  .x.  =  (.g
`  G )
5351, 52gsumconst 16532 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  Fin  /\  X  e.  C )  ->  ( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5447, 4, 49, 53syl3anc 1219 . . . 4  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5554adantr 465 . . 3  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5624, 42, 553eqtrd 2496 . 2  |-  ( ( ( ( R  e. 
CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C ) )  /\  A. i  e.  N  A. j  e.  N  (
i M j )  =  if ( i  =  j ,  X ,  .0.  ) )  -> 
( D `  M
)  =  ( (
# `  N )  .x.  X ) )
5756ex 434 1  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( A. i  e.  N  A. j  e.  N  ( i M j )  =  if ( i  =  j ,  X ,  .0.  )  ->  ( D `  M
)  =  ( (
# `  N )  .x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   ifcif 3889    |-> cmpt 4448   ` cfv 5516  (class class class)co 6190   Fincfn 7410   #chash 12204   Basecbs 14276   0gc0g 14480    gsumg cgsu 14481   Mndcmnd 15511  .gcmg 15516  CMndccmn 16381  mulGrpcmgp 16696   CRingccrg 16752   Mat cmat 18389   maDet cmdat 18506
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  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 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-rp 11093  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-word 12331  df-concat 12333  df-s1 12334  df-substr 12335  df-splice 12336  df-reverse 12337  df-s2 12577  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-pws 14490  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-mulg 15650  df-subg 15780  df-ghm 15847  df-gim 15889  df-cntz 15937  df-oppg 15963  df-symg 15985  df-pmtr 16050  df-psgn 16099  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-dvr 16881  df-rnghom 16912  df-drng 16940  df-subrg 16969  df-sra 17359  df-rgmod 17360  df-cnfld 17928  df-zring 17993  df-zrh 18044  df-dsmm 18266  df-frlm 18281  df-mat 18391  df-mdet 18507
This theorem is referenced by:  mdet1  18523
  Copyright terms: Public domain W3C validator