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.

Step	Hyp	Ref	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 )
9	1, 2, 3, 4, 5, 6, 7, 8	mdetfval 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 )
11	10	biimpi 194	. . . . . 6 |- ( N e/ Fin -> -. N e. Fin )
12	11	intnanrd 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 ) ) = (/) )
14	12, 13	syl 16	. . . 4 |- ( N e/ Fin -> ( Base ` ( N Mat R ) ) = (/) )
15	14	mpteq1d 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 ) ) ) ) ) ) ) = (/)
17	15, 16	syl6eq 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 ) ) ) ) ) ) ) = (/) )
18	9, 17	syl5eq 2507	1 |- ( N e/ Fin -> D = (/) )

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   gsum_g 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