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Theorem mdetleib1 18515
Description: Full substitution of an alternative determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by AV, 26-Dec-2018.)
Hypotheses
Ref Expression
mdetfval1.d  |-  D  =  ( N maDet  R )
mdetfval1.a  |-  A  =  ( N Mat  R )
mdetfval1.b  |-  B  =  ( Base `  A
)
mdetfval1.p  |-  P  =  ( Base `  ( SymGrp `
 N ) )
mdetfval1.y  |-  Y  =  ( ZRHom `  R
)
mdetfval1.s  |-  S  =  (pmSgn `  N )
mdetfval1.t  |-  .x.  =  ( .r `  R )
mdetfval1.u  |-  U  =  (mulGrp `  R )
Assertion
Ref Expression
mdetleib1  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
Distinct variable groups:    B, p    x, p, N    P, p    R, p, x    M, p, x
Allowed substitution hints:    A( x, p)    B( x)    D( x, p)    P( x)    S( x, p)    .x. ( x, p)    U( x, p)    Y( x, p)

Proof of Theorem mdetleib1
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 oveq 6198 . . . . . . 7  |-  ( m  =  M  ->  (
( p `  x
) m x )  =  ( ( p `
 x ) M x ) )
21mpteq2dv 4479 . . . . . 6  |-  ( m  =  M  ->  (
x  e.  N  |->  ( ( p `  x
) m x ) )  =  ( x  e.  N  |->  ( ( p `  x ) M x ) ) )
32oveq2d 6208 . . . . 5  |-  ( m  =  M  ->  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) )
43oveq2d 6208 . . . 4  |-  ( m  =  M  ->  (
( Y `  ( S `  p )
)  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x
) m x ) ) ) )  =  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) )
54mpteq2dv 4479 . . 3  |-  ( m  =  M  ->  (
p  e.  P  |->  ( ( Y `  ( S `  p )
)  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x
) m x ) ) ) ) )  =  ( p  e.  P  |->  ( ( Y `
 ( S `  p ) )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )
65oveq2d 6208 . 2  |-  ( m  =  M  ->  ( 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 ) ) ) ) ) ) )
7 mdetfval1.d . . 3  |-  D  =  ( N maDet  R )
8 mdetfval1.a . . 3  |-  A  =  ( N Mat  R )
9 mdetfval1.b . . 3  |-  B  =  ( Base `  A
)
10 mdetfval1.p . . 3  |-  P  =  ( Base `  ( SymGrp `
 N ) )
11 mdetfval1.y . . 3  |-  Y  =  ( ZRHom `  R
)
12 mdetfval1.s . . 3  |-  S  =  (pmSgn `  N )
13 mdetfval1.t . . 3  |-  .x.  =  ( .r `  R )
14 mdetfval1.u . . 3  |-  U  =  (mulGrp `  R )
157, 8, 9, 10, 11, 12, 13, 14mdetfval1 18514 . 2  |-  D  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
16 ovex 6217 . 2  |-  ( R 
gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )  e.  _V
176, 15, 16fvmpt 5875 1  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   Basecbs 14278   .rcmulr 14343    gsumg cgsu 14483   SymGrpcsymg 15986  pmSgncpsgn 16099  mulGrpcmgp 16698   ZRHomczrh 18042   Mat cmat 18391   maDet cmdat 18508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-word 12333  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-tset 14361  df-symg 15987  df-psgn 16101  df-mat 18393  df-mdet 18509
This theorem is referenced by:  m2detleib  18555
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