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Theorem nfimdetndef 18530
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 2454 . . 3  |-  ( N Mat 
R )  =  ( N Mat  R )
3 eqid 2454 . . 3  |-  ( Base `  ( N Mat  R ) )  =  ( Base `  ( N Mat  R ) )
4 eqid 2454 . . 3  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2454 . . 3  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2454 . . 3  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 eqid 2454 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2454 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetfval 18527 . 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 2651 . . . . . . 7  |-  ( N  e/  Fin  <->  -.  N  e.  Fin )
1110biimpi 194 . . . . . 6  |-  ( N  e/  Fin  ->  -.  N  e.  Fin )
1211intnanrd 908 . . . . 5  |-  ( N  e/  Fin  ->  -.  ( N  e.  Fin  /\  R  e.  _V )
)
13 matbas0 18415 . . . . 5  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
1412, 13syl 16 . . . 4  |-  ( N  e/  Fin  ->  ( Base `  ( N Mat  R
) )  =  (/) )
1514mpteq1d 4484 . . 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 5649 . . 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 2511 . 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 2507 1  |-  ( N  e/  Fin  ->  D  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    e/ wnel 2649   _Vcvv 3078   (/)c0 3748    |-> cmpt 4461    o. ccom 4955   ` cfv 5529  (class class class)co 6203   Fincfn 7423   Basecbs 14295   .rcmulr 14361    gsumg cgsu 14501   SymGrpcsymg 16004  pmSgncpsgn 16117  mulGrpcmgp 16716   ZRHomczrh 18059   Mat cmat 18408   maDet cmdat 18525
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-slot 14299  df-base 14300  df-mat 18410  df-mdet 18526
This theorem is referenced by:  mdetfval1  18531
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