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Theorem mdetdiagid 19673
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 463 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  R  e.  CRing )
21adantr 471 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  R  e.  CRing )
3 simpr 467 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  N  e.  Fin )
43adantr 471 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  N  e.  Fin )
5 simpl 463 . . . . . . 7  |-  ( ( M  e.  B  /\  X  e.  C )  ->  M  e.  B )
65adantl 472 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  M  e.  B )
72, 4, 63jca 1194 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( R  e.  CRing  /\  N  e.  Fin  /\  M  e.  B ) )
87adantr 471 . . . 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 3904 . . . . . . . . . . 11  |-  ( i  =/=  j  ->  if ( i  =  j ,  X ,  .0.  )  =  .0.  )
1110adantl 472 . . . . . . . . . 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 2517 . . . . . . . . 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 613 . . . . . . . 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 81 . . . . . . 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 2807 . . . . . 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 2807 . . . . 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 435 . . . 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 19672 . . . 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 62 . . 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 6321 . . . . . . . . . . . 12  |-  ( i  =  k  ->  (
i M j )  =  ( k M j ) )
26 equequ1 1877 . . . . . . . . . . . . 13  |-  ( i  =  k  ->  (
i  =  j  <->  k  =  j ) )
2726ifbid 3914 . . . . . . . . . . . 12  |-  ( i  =  k  ->  if ( i  =  j ,  X ,  .0.  )  =  if (
k  =  j ,  X ,  .0.  )
)
2825, 27eqeq12d 2476 . . . . . . . . . . 11  |-  ( i  =  k  ->  (
( i M j )  =  if ( i  =  j ,  X ,  .0.  )  <->  ( k M j )  =  if ( k  =  j ,  X ,  .0.  ) ) )
29 oveq2 6322 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
k M j )  =  ( k M k ) )
30 equequ2 1878 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
k  =  j  <->  k  =  k ) )
3130ifbid 3914 . . . . . . . . . . . 12  |-  ( j  =  k  ->  if ( k  =  j ,  X ,  .0.  )  =  if (
k  =  k ,  X ,  .0.  )
)
3229, 31eqeq12d 2476 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
( k M j )  =  if ( k  =  j ,  X ,  .0.  )  <->  ( k M k )  =  if ( k  =  k ,  X ,  .0.  ) ) )
3328, 32rspc2v 3170 . . . . . . . . . 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 655 . . . . . . . . 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 472 . . . . . . . 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 435 . . . . . . 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 1865 . . . . . . . 8  |-  k  =  k
3837iftruei 3899 . . . . . . 7  |-  if ( k  =  k ,  X ,  .0.  )  =  X
3936, 38syl6eq 2511 . . . . . 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 818 . . . . 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 4502 . . . 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 6330 . . 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 17836 . . . . . . . 8  |-  ( R  e.  CRing  ->  G  e. CMnd )
44 cmnmnd 17493 . . . . . . . 8  |-  ( G  e. CMnd  ->  G  e.  Mnd )
4543, 44syl 17 . . . . . . 7  |-  ( R  e.  CRing  ->  G  e.  Mnd )
4645adantr 471 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  G  e.  Mnd )
4746adantr 471 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  G  e.  Mnd )
48 simpr 467 . . . . . 6  |-  ( ( M  e.  B  /\  X  e.  C )  ->  X  e.  C )
4948adantl 472 . . . . 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 17777 . . . . . 6  |-  C  =  ( Base `  G
)
52 mdetdiagid.t . . . . . 6  |-  .x.  =  (.g
`  G )
5351, 52gsumconst 17615 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  Fin  /\  X  e.  C )  ->  ( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5447, 4, 49, 53syl3anc 1276 . . . 4  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5554adantr 471 . . 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 2499 . 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 440 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 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   ifcif 3892    |-> cmpt 4474   ` cfv 5600  (class class class)co 6314   Fincfn 7594   #chash 12546   Basecbs 15169   0gc0g 15386    gsumg cgsu 15387   Mndcmnd 16583  .gcmg 16720  CMndccmn 17478  mulGrpcmgp 17771   CRingccrg 17829   Mat cmat 19480   maDet cmdat 19657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-addf 9643  ax-mulf 9644
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-xor 1416  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-ot 3988  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-er 7388  df-map 7499  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-sup 7981  df-oi 8050  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-rp 11331  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-word 12696  df-lsw 12697  df-concat 12698  df-s1 12699  df-substr 12700  df-splice 12701  df-reverse 12702  df-s2 12980  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-starv 15253  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-unif 15261  df-hom 15262  df-cco 15263  df-0g 15388  df-gsum 15389  df-prds 15394  df-pws 15396  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-submnd 16631  df-grp 16721  df-minusg 16722  df-mulg 16724  df-subg 16862  df-ghm 16929  df-gim 16971  df-cntz 17019  df-oppg 17045  df-symg 17067  df-pmtr 17131  df-psgn 17180  df-cmn 17480  df-abl 17481  df-mgp 17772  df-ur 17784  df-ring 17830  df-cring 17831  df-oppr 17899  df-dvdsr 17917  df-unit 17918  df-invr 17948  df-dvr 17959  df-rnghom 17991  df-drng 18025  df-subrg 18054  df-sra 18443  df-rgmod 18444  df-cnfld 19019  df-zring 19088  df-zrh 19123  df-dsmm 19343  df-frlm 19358  df-mat 19481  df-mdet 19658
This theorem is referenced by:  mdet1  19674
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