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

Theorem nfimdetndef 19273
Description: The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
Hypothesis
Ref Expression
nfimdetndef.d  |-  D  =  ( N maDet  R )
Assertion
Ref Expression
nfimdetndef  |-  ( N  e/  Fin  ->  D  =  (/) )

Proof of Theorem nfimdetndef
Dummy variables  m  p  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfimdetndef.d . . 3  |-  D  =  ( N maDet  R )
2 eqid 2400 . . 3  |-  ( N Mat 
R )  =  ( N Mat  R )
3 eqid 2400 . . 3  |-  ( Base `  ( N Mat  R ) )  =  ( Base `  ( N Mat  R ) )
4 eqid 2400 . . 3  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2400 . . 3  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2400 . . 3  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 eqid 2400 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2400 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetfval 19270 . 2  |-  D  =  ( 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 ) ) ) ) ) ) )
10 df-nel 2599 . . . . . . 7  |-  ( N  e/  Fin  <->  -.  N  e.  Fin )
1110biimpi 194 . . . . . 6  |-  ( N  e/  Fin  ->  -.  N  e.  Fin )
1211intnanrd 916 . . . . 5  |-  ( N  e/  Fin  ->  -.  ( N  e.  Fin  /\  R  e.  _V )
)
13 matbas0 19094 . . . . 5  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
1412, 13syl 17 . . . 4  |-  ( N  e/  Fin  ->  ( Base `  ( N Mat  R
) )  =  (/) )
1514mpteq1d 4473 . . 3  |-  ( N  e/  Fin  ->  (
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.  (/)  |->  ( 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 ) ) ) ) ) ) ) )
16 mpt0 5645 . . 3  |-  ( m  e.  (/)  |->  ( 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 ) ) ) ) ) ) )  =  (/)
1715, 16syl6eq 2457 . 2  |-  ( N  e/  Fin  ->  (
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 ) ) ) ) ) ) )  =  (/) )
189, 17syl5eq 2453 1  |-  ( N  e/  Fin  ->  D  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    e/ wnel 2597   _Vcvv 3056   (/)c0 3735    |-> cmpt 4450    o. ccom 4944   ` cfv 5523  (class class class)co 6232   Fincfn 7472   Basecbs 14731   .rcmulr 14800    gsumg cgsu 14945   SymGrpcsymg 16616  pmSgncpsgn 16728  mulGrpcmgp 17351   ZRHomczrh 18727   Mat cmat 19091   maDet cmdat 19268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  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 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-slot 14735  df-base 14736  df-mat 19092  df-mdet 19269
This theorem is referenced by:  mdetfval1  19274
  Copyright terms: Public domain W3C validator