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Theorem mdetfval 27590
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 5883 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
3 mdetfval.a . . . . . . . 8  |-  A  =  ( N Mat  R )
42, 3syl6eqr 2346 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
54fveq2d 5545 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
6 mdetfval.b . . . . . 6  |-  B  =  ( Base `  A
)
75, 6syl6eqr 2346 . . . . 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 5545 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( SymGrp `  n )  =  ( SymGrp `  N
) )
1110fveq2d 5545 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  ( Base `  ( SymGrp `
 N ) ) )
12 mdetfval.p . . . . . . . 8  |-  P  =  ( Base `  ( SymGrp `
 N ) )
1311, 12syl6eqr 2346 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  ( SymGrp `
 n ) )  =  P )
14 fveq2 5541 . . . . . . . . . 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 2346 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .r `  r
)  =  .x.  )
188fveq2d 5545 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  ( ZRHom `  R ) )
19 mdetfval.y . . . . . . . . . 10  |-  Y  =  ( ZRHom `  R
)
2018, 19syl6eqr 2346 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ZRHom `  r
)  =  Y )
219fveq2d 5545 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  (pmSgn `  N )
)
22 mdetfval.s . . . . . . . . . . 11  |-  S  =  (pmSgn `  N )
2321, 22syl6eqr 2346 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (pmSgn `  n )  =  S )
2423fveq1d 5543 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( (pmSgn `  n
) `  p )  =  ( S `  p ) )
2520, 24fveq12d 5547 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( ZRHom `  r ) `  (
(pmSgn `  n ) `  p ) )  =  ( Y `  ( S `  p )
) )
268fveq2d 5545 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  (mulGrp `  R )
)
27 mdetfval.u . . . . . . . . . 10  |-  U  =  (mulGrp `  R )
2826, 27syl6eqr 2346 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  (mulGrp `  r )  =  U )
29 eqidd 2297 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( p `  x ) m x )  =  ( ( p `  x ) m x ) )
309, 29mpteq12dv 4114 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( x  e.  n  |->  ( ( p `  x ) m x ) )  =  ( x  e.  N  |->  ( ( p `  x
) m x ) ) )
3128, 30oveq12d 5892 . . . . . . . 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 5895 . . . . . . 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 4114 . . . . . 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 5892 . . . . 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 4114 . . . 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 27588 . . . 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 5555 . . . . . 6  |-  ( Base `  A )  e.  _V
386, 37eqeltri 2366 . . . . 5  |-  B  e. 
_V
3938mptex 5762 . . . 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 5994 . . 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 5971 . . . . . 6  |-  Rel  dom maDet
4241ovprc 5901 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maDet  R )  =  (/) )
43 mpt0 5387 . . . . 5  |-  ( m  e.  (/)  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )  =  (/)
4442, 43syl6eqr 2346 . . . 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 27545 . . . . . . . . . 10  |- Mat  =  ( y  e.  Fin , 
z  e.  _V  |->  ( ( z freeLMod  ( y  X.  y ) ) sSet  <. ( .r `  ndx ) ,  ( z maMul  <.
y ,  y ,  y >. ) >. )
)
4645reldmmpt2 5971 . . . . . . . . 9  |-  Rel  dom Mat
4746ovprc 5901 . . . . . . . 8  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
483, 47syl5eq 2340 . . . . . . 7  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  A  =  (/) )
4948fveq2d 5545 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
50 base0 13201 . . . . . 6  |-  (/)  =  (
Base `  (/) )
5149, 6, 503eqtr4g 2353 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
52 eqidd 2297 . . . . 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 4114 . . . 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 2331 . . 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 2316 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 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656   <.cotp 3657    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874   Fincfn 6879   ndxcnx 13161   sSet csts 13162   Basecbs 13164   .rcmulr 13225    gsumg cgsu 13417   SymGrpcsymg 14785  mulGrpcmgp 15341   ZRHomczrh 16467   freeLMod cfrlm 27315  pmSgncpsgn 27517   maMul cmmul 27542   Mat cmat 27543   maDet cmdat 27586
This theorem is referenced by:  mdetleib  27591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-slot 13168  df-base 13169  df-mat 27545  df-mdet 27588
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