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

Theorem mdetfval 18957
Description: First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)
Hypotheses
Ref Expression
mdetfval.d  |-  D  =  ( N maDet  R )
mdetfval.a  |-  A  =  ( N Mat  R )
mdetfval.b  |-  B  =  ( Base `  A
)
mdetfval.p  |-  P  =  ( Base `  ( SymGrp `
 N ) )
mdetfval.y  |-  Y  =  ( ZRHom `  R
)
mdetfval.s  |-  S  =  (pmSgn `  N )
mdetfval.t  |-  .x.  =  ( .r `  R )
mdetfval.u  |-  U  =  (mulGrp `  R )
Assertion
Ref Expression
mdetfval  |-  D  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
Distinct variable groups:    B, m    m, p, x, N    P, m    R, m, p, x    S, m    .x. , m    U, m    m, Y
Allowed substitution hints:    A( x, m, p)    B( x, p)    D( x, m, p)    P( x, p)    S( x, p)    .x. ( x, p)    U( x, p)    Y( x, p)

Proof of Theorem mdetfval
Dummy variables  y 
z  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetfval.d . 2  |-  D  =  ( N maDet  R )
2 oveq12 6304 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 mdetfval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2526 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5876 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 mdetfval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2526 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpr 461 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  r  =  R )
9 simpl 457 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
109fveq2d 5876 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( SymGrp `  n )  =  ( SymGrp `  N
) )
1110fveq2d 5876 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  ( Base `  ( SymGrp `
 N ) ) )
12 mdetfval.p . . . . . . . 8  |-  P  =  ( Base `  ( SymGrp `
 N ) )
1311, 12syl6eqr 2526 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  P )
14 fveq2 5872 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
1514adantl 466 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  ( .r
`  R ) )
16 mdetfval.t . . . . . . . . 9  |-  .x.  =  ( .r `  R )
1715, 16syl6eqr 2526 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  .x.  )
188fveq2d 5876 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  ( ZRHom `  R ) )
19 mdetfval.y . . . . . . . . . . 11  |-  Y  =  ( ZRHom `  R
)
2018, 19syl6eqr 2526 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  Y )
21 fveq2 5872 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (pmSgn `  n )  =  (pmSgn `  N ) )
2221adantr 465 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  (pmSgn `  N )
)
23 mdetfval.s . . . . . . . . . . 11  |-  S  =  (pmSgn `  N )
2422, 23syl6eqr 2526 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  S )
2520, 24coeq12d 5173 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ZRHom `  r )  o.  (pmSgn `  n ) )  =  ( Y  o.  S
) )
2625fveq1d 5874 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ( ZRHom `  r )  o.  (pmSgn `  n ) ) `  p )  =  ( ( Y  o.  S
) `  p )
)
27 fveq2 5872 . . . . . . . . . . 11  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
2827adantl 466 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  (mulGrp `  R )
)
29 mdetfval.u . . . . . . . . . 10  |-  U  =  (mulGrp `  R )
3028, 29syl6eqr 2526 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  U )
319mpteq1d 4534 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( x  e.  n  |->  ( ( p `  x ) m x ) )  =  ( x  e.  N  |->  ( ( p `  x
) m x ) ) )
3230, 31oveq12d 6313 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) )
3317, 26, 32oveq123d 6316 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ( ( ZRHom `  r )  o.  (pmSgn `  n )
) `  p )
( .r `  r
) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) )  =  ( ( ( Y  o.  S ) `
 p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) )
3413, 33mpteq12dv 4531 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( p  e.  (
Base `  ( SymGrp `  n ) )  |->  ( ( ( ( ZRHom `  r )  o.  (pmSgn `  n ) ) `  p ) ( .r
`  r ) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x
) m x ) ) ) ) )  =  ( p  e.  P  |->  ( ( ( Y  o.  S ) `
 p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) )
358, 34oveq12d 6313 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r  gsumg  ( p  e.  (
Base `  ( SymGrp `  n ) )  |->  ( ( ( ( ZRHom `  r )  o.  (pmSgn `  n ) ) `  p ) ( .r
`  r ) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x
) m x ) ) ) ) ) )  =  ( R 
gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
367, 35mpteq12dv 4531 . . . 4  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  e.  (
Base `  ( n Mat  r ) )  |->  ( r  gsumg  ( p  e.  (
Base `  ( SymGrp `  n ) )  |->  ( ( ( ( ZRHom `  r )  o.  (pmSgn `  n ) ) `  p ) ( .r
`  r ) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x
) m x ) ) ) ) ) ) )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
37 df-mdet 18956 . . . 4  |- maDet  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( r  gsumg  ( p  e.  ( Base `  ( SymGrp `
 n ) ) 
|->  ( ( ( ( ZRHom `  r )  o.  (pmSgn `  n )
) `  p )
( .r `  r
) ( (mulGrp `  r )  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
38 fvex 5882 . . . . . 6  |-  ( Base `  A )  e.  _V
396, 38eqeltri 2551 . . . . 5  |-  B  e. 
_V
4039mptex 6142 . . . 4  |-  ( m  e.  B  |->  ( R 
gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  e. 
_V
4136, 37, 40ovmpt2a 6428 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S
) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
4237reldmmpt2 6408 . . . . . 6  |-  Rel  dom maDet
4342ovprc 6322 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  (/) )
44 mpt0 5714 . . . . 5  |-  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S
) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  (/)
4543, 44syl6eqr 2526 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S
) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
46 df-mat 18779 . . . . . . . . . 10  |- Mat  =  ( y  e.  Fin , 
z  e.  _V  |->  ( ( z freeLMod  ( y  X.  y ) ) sSet  <. ( .r `  ndx ) ,  ( z maMul  <.
y ,  y ,  y >. ) >. )
)
4746reldmmpt2 6408 . . . . . . . . 9  |-  Rel  dom Mat
4847ovprc 6322 . . . . . . . 8  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
493, 48syl5eq 2520 . . . . . . 7  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  A  =  (/) )
5049fveq2d 5876 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
51 base0 14546 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5250, 6, 513eqtr4g 2533 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
5352mpteq1d 4534 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
5445, 53eqtr4d 2511 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S
) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
5541, 54pm2.61i 164 . 2  |-  ( N maDet 
R )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
561, 55eqtri 2496 1  |-  D  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   <.cop 4039   <.cotp 4041    |-> cmpt 4511    X. cxp 5003    o. ccom 5009   ` cfv 5594  (class class class)co 6295   Fincfn 7528   ndxcnx 14504   sSet csts 14505   Basecbs 14507   .rcmulr 14573    gsumg cgsu 14713   SymGrpcsymg 16274  pmSgncpsgn 16387  mulGrpcmgp 17013   ZRHomczrh 18406   freeLMod cfrlm 18646   maMul cmmul 18754   Mat cmat 18778   maDet cmdat 18955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-slot 14511  df-base 14512  df-mat 18779  df-mdet 18956
This theorem is referenced by:  mdetleib  18958  nfimdetndef  18960  mdetfval1  18961  mdet0pr  18963  mdetf  18966
  Copyright terms: Public domain W3C validator