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Theorem mdetdiagid 19272
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 455 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  R  e.  CRing )
21adantr 463 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  R  e.  CRing )
3 simpr 459 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  N  e.  Fin )
43adantr 463 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  N  e.  Fin )
5 simpl 455 . . . . . . 7  |-  ( ( M  e.  B  /\  X  e.  C )  ->  M  e.  B )
65adantl 464 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  M  e.  B )
72, 4, 63jca 1174 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( R  e.  CRing  /\  N  e.  Fin  /\  M  e.  B ) )
87adantr 463 . . . 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 3941 . . . . . . . . . . 11  |-  ( i  =/=  j  ->  if ( i  =  j ,  X ,  .0.  )  =  .0.  )
1110adantl 464 . . . . . . . . . 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 602 . . . . . . . 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 2862 . . . . . 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 2862 . . . . 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 427 . . . 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 19271 . . . 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 6277 . . . . . . . . . . . 12  |-  ( i  =  k  ->  (
i M j )  =  ( k M j ) )
26 equequ1 1803 . . . . . . . . . . . . 13  |-  ( i  =  k  ->  (
i  =  j  <->  k  =  j ) )
2726ifbid 3951 . . . . . . . . . . . 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 6278 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
k M j )  =  ( k M k ) )
30 equequ2 1804 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
k  =  j  <->  k  =  k ) )
3130ifbid 3951 . . . . . . . . . . . 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 3216 . . . . . . . . . 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 643 . . . . . . . . 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 464 . . . . . . . 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 427 . . . . . . 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 1796 . . . . . . . 8  |-  k  =  k
3837iftruei 3936 . . . . . . 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 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 4525 . . . 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 6286 . . 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 17404 . . . . . . . 8  |-  ( R  e.  CRing  ->  G  e. CMnd )
44 cmnmnd 17015 . . . . . . . 8  |-  ( G  e. CMnd  ->  G  e.  Mnd )
4543, 44syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  G  e.  Mnd )
4645adantr 463 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  G  e.  Mnd )
4746adantr 463 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  G  e.  Mnd )
48 simpr 459 . . . . . 6  |-  ( ( M  e.  B  /\  X  e.  C )  ->  X  e.  C )
4948adantl 464 . . . . 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 17345 . . . . . 6  |-  C  =  ( Base `  G
)
52 mdetdiagid.t . . . . . 6  |-  .x.  =  (.g
`  G )
5351, 52gsumconst 17155 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  Fin  /\  X  e.  C )  ->  ( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5447, 4, 49, 53syl3anc 1226 . . . 4  |-  ( ( ( R  e.  CRing  /\  N  e.  Fin )  /\  ( M  e.  B  /\  X  e.  C
) )  ->  ( G  gsumg  ( k  e.  N  |->  X ) )  =  ( ( # `  N
)  .x.  X )
)
5554adantr 463 . . 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 432 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   ifcif 3929    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   Fincfn 7509   #chash 12390   Basecbs 14719   0gc0g 14932    gsumg cgsu 14933   Mndcmnd 16121  .gcmg 16258  CMndccmn 17000  mulGrpcmgp 17339   CRingccrg 17397   Mat cmat 19079   maDet cmdat 19256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-xor 1363  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-word 12529  df-lsw 12530  df-concat 12531  df-s1 12532  df-substr 12533  df-splice 12534  df-reverse 12535  df-s2 12807  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-0g 14934  df-gsum 14935  df-prds 14940  df-pws 14942  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-grp 16259  df-minusg 16260  df-mulg 16262  df-subg 16400  df-ghm 16467  df-gim 16509  df-cntz 16557  df-oppg 16583  df-symg 16605  df-pmtr 16669  df-psgn 16718  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-cring 17399  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-dvr 17530  df-rnghom 17562  df-drng 17596  df-subrg 17625  df-sra 18016  df-rgmod 18017  df-cnfld 18619  df-zring 18687  df-zrh 18719  df-dsmm 18939  df-frlm 18954  df-mat 19080  df-mdet 19257
This theorem is referenced by:  mdet1  19273
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