Theorem mdetfval 20392

Description: First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)

Hypotheses

Ref	Expression
mdetfval.d	|- D = ( N maDet R )
mdetfval.a	|- A = ( N Mat R )
mdetfval.b	|- B = ( Base ` A )
mdetfval.p	|- P = ( Base ` ( SymGrp ` N ) )
mdetfval.y	|- Y = ( ZRHom ` R )
mdetfval.s	|- S = (pmSgn ` N )
mdetfval.t	|- .x. = ( .r ` R )
mdetfval.u	|- U = (mulGrp ` R )

Assertion

Ref	Expression
mdetfval	|- D = ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) )

Distinct variable groups:   B, m   m, p, x, N   P, m   R, m, p, x   S, m   .x. , m   U, m   m, Y

Allowed substitution hints:   A( x, m, p)   B( x, p)   D( x, m, p)   P( x, p)   S( x, p)   .x. ( x, p)   U( x, p)   Y( x, p)

Proof of Theorem mdetfval

Dummy variables y z n r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdetfval.d	. 2 |- D = ( N maDet R )
2		oveq12 6659	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
3		mdetfval.a	. . . . . . . 8 |- A = ( N Mat R )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( n Mat r ) = A )
5	4	fveq2d 6195	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` A ) )
6		mdetfval.b	. . . . . 6 |- B = ( Base ` A )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
8		simpr 477	. . . . . 6 |- ( ( n = N /\ r = R ) -> r = R )
9		simpl 473	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> n = N )
10	9	fveq2d 6195	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( SymGrp ` n ) = ( SymGrp ` N ) )
11	10	fveq2d 6195	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( Base ` ( SymGrp ` n ) ) = ( Base ` ( SymGrp ` N ) ) )
12		mdetfval.p	. . . . . . . 8 |- P = ( Base ` ( SymGrp ` N ) )
13	11, 12	syl6eqr 2674	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( Base ` ( SymGrp ` n ) ) = P )
14		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
15	14	adantl 482	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( .r ` r ) = ( .r ` R ) )
16		mdetfval.t	. . . . . . . . 9 |- .x. = ( .r ` R )
17	15, 16	syl6eqr 2674	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( .r ` r ) = .x. )
18	8	fveq2d 6195	. . . . . . . . . . 11 |- ( ( n = N /\ r = R ) -> ( ZRHom ` r ) = ( ZRHom ` R ) )
19		mdetfval.y	. . . . . . . . . . 11 |- Y = ( ZRHom ` R )
20	18, 19	syl6eqr 2674	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> ( ZRHom ` r ) = Y )
21		fveq2 6191	. . . . . . . . . . . 12 |- ( n = N -> (pmSgn ` n ) = (pmSgn ` N ) )
22	21	adantr 481	. . . . . . . . . . 11 |- ( ( n = N /\ r = R ) -> (pmSgn ` n ) = (pmSgn ` N ) )
23		mdetfval.s	. . . . . . . . . . 11 |- S = (pmSgn ` N )
24	22, 23	syl6eqr 2674	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> (pmSgn ` n ) = S )
25	20, 24	coeq12d 5286	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( ( ZRHom ` r ) o. (pmSgn ` n ) ) = ( Y o. S ) )
26	25	fveq1d 6193	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( ( ( ZRHom ` r ) o. (pmSgn ` n ) ) ` p ) = ( ( Y o. S ) ` p ) )
27		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
28	27	adantl 482	. . . . . . . . . 10 |- ( ( n = N /\ r = R ) -> (mulGrp ` r ) = (mulGrp ` R ) )
29		mdetfval.u	. . . . . . . . . 10 |- U = (mulGrp ` R )
30	28, 29	syl6eqr 2674	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> (mulGrp ` r ) = U )
31	9	mpteq1d 4738	. . . . . . . . 9 |- ( ( n = N /\ r = R ) -> ( x e. n |-> ( ( p ` x ) m x ) ) = ( x e. N |-> ( ( p ` x ) m x ) ) )
32	30, 31	oveq12d 6668	. . . . . . . 8 |- ( ( n = N /\ r = R ) -> ( (mulGrp ` r ) gsumg ( x e. n |-> ( ( p ` x ) m x ) ) ) = ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) )
33	17, 26, 32	oveq123d 6671	. . . . . . 7 |- ( ( n = N /\ r = R ) -> ( ( ( ( ZRHom ` r ) o. (pmSgn ` n ) ) ` p ) ( .r ` r ) ( (mulGrp ` r ) gsumg ( x e. n |-> ( ( p ` x ) m x ) ) ) ) = ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) )
34	13, 33	mpteq12dv 4733	. . . . . 6 |- ( ( n = N /\ r = R ) -> ( p e. ( Base ` ( SymGrp ` n ) ) |-> ( ( ( ( ZRHom ` r ) o. (pmSgn ` n ) ) ` p ) ( .r ` r ) ( (mulGrp ` r ) gsumg ( x e. n |-> ( ( p ` x ) m x ) ) ) ) ) = ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) )
35	8, 34	oveq12d 6668	. . . . 5 |- ( ( n = N /\ r = 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 ) ) ) ) ) ) = ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) )
36	7, 35	mpteq12dv 4733	. . . 4 |- ( ( n = N /\ r = R ) -> ( 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. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) )
37		df-mdet 20391	. . . 4 |- maDet = ( n e. _V , r e. _V |-> ( 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 ) ) ) ) ) ) ) )
38		fvex 6201	. . . . . 6 |- ( Base ` A ) e. _V
39	6, 38	eqeltri 2697	. . . . 5 |- B e. _V
40	39	mptex 6486	. . . 4 |- ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) e. _V
41	36, 37, 40	ovmpt2a 6791	. . 3 |- ( ( N e. _V /\ R e. _V ) -> ( N maDet R ) = ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) )
42	37	reldmmpt2 6771	. . . . . 6 |- Rel dom maDet
43	42	ovprc 6683	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( N maDet R ) = (/) )
44		mpt0 6021	. . . . 5 |- ( m e. (/) |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) = (/)
45	43, 44	syl6eqr 2674	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( N maDet R ) = ( m e. (/) |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) )
46		df-mat 20214	. . . . . . . . . 10 |- Mat = ( y e. Fin , z e. _V |-> ( ( z freeLMod ( y X. y ) ) sSet <. ( .r ` ndx ) , ( z maMul <. y , y , y >. ) >. ) )
47	46	reldmmpt2 6771	. . . . . . . . 9 |- Rel dom Mat
48	47	ovprc 6683	. . . . . . . 8 |- ( -. ( N e. _V /\ R e. _V ) -> ( N Mat R ) = (/) )
49	3, 48	syl5eq 2668	. . . . . . 7 |- ( -. ( N e. _V /\ R e. _V ) -> A = (/) )
50	49	fveq2d 6195	. . . . . 6 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` A ) = ( Base ` (/) ) )
51		base0 15912	. . . . . 6 |- (/) = ( Base ` (/) )
52	50, 6, 51	3eqtr4g 2681	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
53	52	mpteq1d 4738	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) = ( m e. (/) |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) )
54	45, 53	eqtr4d 2659	. . 3 |- ( -. ( N e. _V /\ R e. _V ) -> ( N maDet R ) = ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) ) )
55	41, 54	pm2.61i 176	. 2 |- ( N maDet R ) = ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) )
56	1, 55	eqtri 2644	1 |- D = ( m e. B |-> ( R gsumg ( p e. P |-> ( ( ( Y o. S ) ` p ) .x. ( U gsumg ( x e. N |-> ( ( p ` x ) m x ) ) ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Fincfn 7955   ndxcnx 15854   sSet csts 15855   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   SymGrpcsymg 17797  pmSgncpsgn 17909  mulGrpcmgp 18489   ZRHomczrh 19848   freeLMod cfrlm 20090   maMul cmmul 20189   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-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:  mdetleib  20393  nfimdetndef  20395  mdetfval1  20396  mdet0pr  20398  mdetf  20401