Theorem nfimdetndef 20395

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 2622	. . 3 |- ( N Mat R ) = ( N Mat R )
3		eqid 2622	. . 3 |- ( Base ` ( N Mat R ) ) = ( Base ` ( N Mat R ) )
4		eqid 2622	. . 3 |- ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
5		eqid 2622	. . 3 |- ( ZRHom ` R ) = ( ZRHom ` R )
6		eqid 2622	. . 3 |- (pmSgn ` N ) = (pmSgn ` N )
7		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
8		eqid 2622	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetfval 20392	. 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 2898	. . . . . . 7 |- ( N e/ Fin <-> -. N e. Fin )
11	10	biimpi 206	. . . . . 6 |- ( N e/ Fin -> -. N e. Fin )
12	11	intnanrd 963	. . . . 5 |- ( N e/ Fin -> -. ( N e. Fin /\ R e. _V ) )
13		matbas0 20216	. . . . 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 4738	. . 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 6021	. . 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 2672	. 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 2668	1 |- ( N e/ Fin -> D = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   SymGrpcsymg 17797  pmSgncpsgn 17909  mulGrpcmgp 18489   ZRHomczrh 19848   Mat cmat 20213   maDet cmdat 20390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-slot 15861  df-base 15863  df-mat 20214  df-mdet 20391

This theorem is referenced by:  mdetfval1  20396