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Theorem mdetfval 27487
Description: First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.)
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 `  ( 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 5867 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 mdetfval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2333 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5529 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 mdetfval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2333 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpr 447 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  r  =  R )
9 simpl 443 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
109fveq2d 5529 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( SymGrp `  n )  =  ( SymGrp `  N
) )
1110fveq2d 5529 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  ( Base `  ( SymGrp `
 N ) ) )
12 mdetfval.p . . . . . . . 8  |-  P  =  ( Base `  ( SymGrp `
 N ) )
1311, 12syl6eqr 2333 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  P )
14 fveq2 5525 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
1514adantl 452 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  ( .r
`  R ) )
16 mdetfval.t . . . . . . . . 9  |-  .x.  =  ( .r `  R )
1715, 16syl6eqr 2333 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  .x.  )
188fveq2d 5529 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  ( ZRHom `  R ) )
19 mdetfval.y . . . . . . . . . 10  |-  Y  =  ( ZRHom `  R
)
2018, 19syl6eqr 2333 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  Y )
219fveq2d 5529 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  (pmSgn `  N )
)
22 mdetfval.s . . . . . . . . . . 11  |-  S  =  (pmSgn `  N )
2321, 22syl6eqr 2333 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  S )
2423fveq1d 5527 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( (pmSgn `  n
) `  p )  =  ( S `  p ) )
2520, 24fveq12d 5531 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) )  =  ( Y `  ( S `  p )
) )
268fveq2d 5529 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  (mulGrp `  R )
)
27 mdetfval.u . . . . . . . . . 10  |-  U  =  (mulGrp `  R )
2826, 27syl6eqr 2333 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  U )
29 eqidd 2284 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( p `  x ) m x )  =  ( ( p `  x ) m x ) )
309, 29mpteq12dv 4098 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( x  e.  n  |->  ( ( p `  x ) m x ) )  =  ( x  e.  N  |->  ( ( p `  x
) m x ) ) )
3128, 30oveq12d 5876 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) )
3217, 25, 31oveq123d 5879 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) ) ( .r `  r ) ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) )  =  ( ( Y `
 ( S `  p ) )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) )
3313, 32mpteq12dv 4098 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( p  e.  (
Base `  ( SymGrp `  n ) )  |->  ( ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) ) ( .r `  r ) ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) )  =  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) )
348, 33oveq12d 5876 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( r  gsumg  ( p  e.  (
Base `  ( SymGrp `  n ) )  |->  ( ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) ) ( .r `  r ) ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) ) )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
357, 34mpteq12dv 4098 . . . 4  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  e.  (
Base `  ( n Mat  r ) )  |->  ( r  gsumg  ( p  e.  (
Base `  ( SymGrp `  n ) )  |->  ( ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) ) ( .r `  r ) ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
36 df-mdet 27485 . . . 4  |- maDet  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( r  gsumg  ( p  e.  ( Base `  ( SymGrp `
 n ) ) 
|->  ( ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) ) ( .r `  r ) ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
37 fvex 5539 . . . . . 6  |-  ( Base `  A )  e.  _V
386, 37eqeltri 2353 . . . . 5  |-  B  e. 
_V
3938mptex 5746 . . . 4  |-  ( m  e.  B  |->  ( R 
gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  e. 
_V
4035, 36, 39ovmpt2a 5978 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
4136reldmmpt2 5955 . . . . . 6  |-  Rel  dom maDet
4241ovprc 5885 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  (/) )
43 mpt0 5371 . . . . 5  |-  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  (/)
4442, 43syl6eqr 2333 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
45 df-mat 27442 . . . . . . . . . 10  |- Mat  =  ( y  e.  Fin , 
z  e.  _V  |->  ( ( z freeLMod  ( y  X.  y ) ) sSet  <. ( .r `  ndx ) ,  ( z maMul  <.
y ,  y ,  y >. ) >. )
)
4645reldmmpt2 5955 . . . . . . . . 9  |-  Rel  dom Mat
4746ovprc 5885 . . . . . . . 8  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
483, 47syl5eq 2327 . . . . . . 7  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  A  =  (/) )
4948fveq2d 5529 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
50 base0 13185 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5149, 6, 503eqtr4g 2340 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
52 eqidd 2284 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
5351, 52mpteq12dv 4098 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
5444, 53eqtr4d 2318 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) ) )
5540, 54pm2.61i 156 . 2  |-  ( N maDet 
R )  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
561, 55eqtri 2303 1  |-  D  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643   <.cotp 3644    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Fincfn 6863   ndxcnx 13145   sSet csts 13146   Basecbs 13148   .rcmulr 13209    gsumg cgsu 13401   SymGrpcsymg 14769  mulGrpcmgp 15325   ZRHomczrh 16451   freeLMod cfrlm 27212  pmSgncpsgn 27414   maMul cmmul 27439   Mat cmat 27440   maDet cmdat 27483
This theorem is referenced by:  mdetleib  27488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-slot 13152  df-base 13153  df-mat 27442  df-mdet 27485
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