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.

Step	Hyp	Ref	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 )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetfval 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 )
11	10	biimpi 194	. . . . . 6 |- ( N e/ Fin -> -. N e. Fin )
12	11	intnanrd 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 ) ) = (/) )
14	12, 13	syl 17	. . . 4 |- ( N e/ Fin -> ( Base ` ( N Mat R ) ) = (/) )
15	14	mpteq1d 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 ) ) ) ) ) ) ) = (/)
17	15, 16	syl6eq 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 ) ) ) ) ) ) ) = (/) )
18	9, 17	syl5eq 2453	1 |- ( N e/ Fin -> D = (/) )

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   gsum_g 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