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Theorem mdetfval 27464
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 6090 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 mdetfval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2486 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5732 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 mdetfval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2486 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
8 simpr 448 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  r  =  R )
9 simpl 444 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
109fveq2d 5732 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( SymGrp `  n )  =  ( SymGrp `  N
) )
1110fveq2d 5732 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  ( Base `  ( SymGrp `
 N ) ) )
12 mdetfval.p . . . . . . . 8  |-  P  =  ( Base `  ( SymGrp `
 N ) )
1311, 12syl6eqr 2486 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  P )
14 fveq2 5728 . . . . . . . . . 10  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
1514adantl 453 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  ( .r
`  R ) )
16 mdetfval.t . . . . . . . . 9  |-  .x.  =  ( .r `  R )
1715, 16syl6eqr 2486 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  .x.  )
188fveq2d 5732 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  ( ZRHom `  R ) )
19 mdetfval.y . . . . . . . . . 10  |-  Y  =  ( ZRHom `  R
)
2018, 19syl6eqr 2486 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  Y )
219fveq2d 5732 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  (pmSgn `  N )
)
22 mdetfval.s . . . . . . . . . . 11  |-  S  =  (pmSgn `  N )
2321, 22syl6eqr 2486 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  S )
2423fveq1d 5730 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( (pmSgn `  n
) `  p )  =  ( S `  p ) )
2520, 24fveq12d 5734 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) )  =  ( Y `  ( S `  p )
) )
268fveq2d 5732 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  (mulGrp `  R )
)
27 mdetfval.u . . . . . . . . . 10  |-  U  =  (mulGrp `  R )
2826, 27syl6eqr 2486 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  U )
299mpteq1d 4290 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( x  e.  n  |->  ( ( p `  x ) m x ) )  =  ( x  e.  N  |->  ( ( p `  x
) m x ) ) )
3028, 29oveq12d 6099 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( (mulGrp `  r
)  gsumg  ( x  e.  n  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) )
3117, 25, 30oveq123d 6102 . . . . . . 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 ) ) ) ) )
3213, 31mpteq12dv 4287 . . . . . 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 ) ) ) ) ) )
338, 32oveq12d 6099 . . . . 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 ) ) ) ) ) ) )
347, 33mpteq12dv 4287 . . . 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 ) ) ) ) ) ) ) )
35 df-mdet 27462 . . . 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 ) ) ) ) ) ) ) )
36 fvex 5742 . . . . . 6  |-  ( Base `  A )  e.  _V
376, 36eqeltri 2506 . . . . 5  |-  B  e. 
_V
3837mptex 5966 . . . 4  |-  ( m  e.  B  |->  ( R 
gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  e. 
_V
3934, 35, 38ovmpt2a 6204 . . 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 ) ) ) ) ) ) ) )
4035reldmmpt2 6181 . . . . . 6  |-  Rel  dom maDet
4140ovprc 6108 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  (/) )
42 mpt0 5572 . . . . 5  |-  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  (/)
4341, 42syl6eqr 2486 . . . 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 ) ) ) ) ) ) ) )
44 df-mat 27419 . . . . . . . . . 10  |- Mat  =  ( y  e.  Fin , 
z  e.  _V  |->  ( ( z freeLMod  ( y  X.  y ) ) sSet  <. ( .r `  ndx ) ,  ( z maMul  <.
y ,  y ,  y >. ) >. )
)
4544reldmmpt2 6181 . . . . . . . . 9  |-  Rel  dom Mat
4645ovprc 6108 . . . . . . . 8  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
473, 46syl5eq 2480 . . . . . . 7  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  A  =  (/) )
4847fveq2d 5732 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
49 base0 13506 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5048, 6, 493eqtr4g 2493 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
5150mpteq1d 4290 . . . 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 ) ) ) ) ) ) ) )
5243, 51eqtr4d 2471 . . 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 ) ) ) ) ) ) ) )
5339, 52pm2.61i 158 . 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 ) ) ) ) ) ) )
541, 53eqtri 2456 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   <.cop 3817   <.cotp 3818    e. cmpt 4266    X. cxp 4876   ` cfv 5454  (class class class)co 6081   Fincfn 7109   ndxcnx 13466   sSet csts 13467   Basecbs 13469   .rcmulr 13530    gsumg cgsu 13724   SymGrpcsymg 15092  mulGrpcmgp 15648   ZRHomczrh 16778   freeLMod cfrlm 27189  pmSgncpsgn 27391   maMul cmmul 27416   Mat cmat 27417   maDet cmdat 27460
This theorem is referenced by:  mdetleib  27465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-slot 13473  df-base 13474  df-mat 27419  df-mdet 27462
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