Theorem madufval 20443

Description: First substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)

Hypotheses

Ref	Expression
madufval.a	|- A = ( N Mat R )
madufval.d	|- D = ( N maDet R )
madufval.j	|- J = ( N maAdju R )
madufval.b	|- B = ( Base ` A )
madufval.o	|- .1. = ( 1r ` R )
madufval.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
madufval	|- J = ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) )

Distinct variable groups:   m, N, i, j, k, l   R, m, i, j, k, l   B, m

Allowed substitution hints:   A( i, j, k, m, l)   B( i, j, k, l)   D( i, j, k, m, l)   .1. ( i, j, k, m, l)   J( i, j, k, m, l)   .0. ( i, j, k, m, l)

Proof of Theorem madufval

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

Step	Hyp	Ref	Expression
1		madufval.j	. 2 |- J = ( N maAdju R )
2		oveq1 6657	. . . . . . 7 |- ( n = N -> ( n Mat r ) = ( N Mat r ) )
3	2	fveq2d 6195	. . . . . 6 |- ( n = N -> ( Base ` ( n Mat r ) ) = ( Base ` ( N Mat r ) ) )
4		id 22	. . . . . . 7 |- ( n = N -> n = N )
5		oveq1 6657	. . . . . . . 8 |- ( n = N -> ( n maDet r ) = ( N maDet r ) )
6		eqidd 2623	. . . . . . . . 9 |- ( n = N -> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) = if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) )
7	4, 4, 6	mpt2eq123dv 6717	. . . . . . . 8 |- ( n = N -> ( k e. n , l e. n |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) = ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) )
8	5, 7	fveq12d 6197	. . . . . . 7 |- ( n = N -> ( ( n maDet r ) ` ( k e. n , l e. n |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) = ( ( N maDet r ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) )
9	4, 4, 8	mpt2eq123dv 6717	. . . . . 6 |- ( n = N -> ( i e. n , j e. n |-> ( ( n maDet r ) ` ( k e. n , l e. n |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) = ( i e. N , j e. N |-> ( ( N maDet r ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) )
10	3, 9	mpteq12dv 4733	. . . . 5 |- ( n = N -> ( m e. ( Base ` ( n Mat r ) ) |-> ( i e. n , j e. n |-> ( ( n maDet r ) ` ( k e. n , l e. n |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) ) = ( m e. ( Base ` ( N Mat r ) ) |-> ( i e. N , j e. N |-> ( ( N maDet r ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) ) )
11		oveq2 6658	. . . . . . 7 |- ( r = R -> ( N Mat r ) = ( N Mat R ) )
12	11	fveq2d 6195	. . . . . 6 |- ( r = R -> ( Base ` ( N Mat r ) ) = ( Base ` ( N Mat R ) ) )
13		oveq2 6658	. . . . . . . 8 |- ( r = R -> ( N maDet r ) = ( N maDet R ) )
14		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
15		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
16	14, 15	ifeq12d 4106	. . . . . . . . . 10 |- ( r = R -> if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) = if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) )
17	16	ifeq1d 4104	. . . . . . . . 9 |- ( r = R -> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) = if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) )
18	17	mpt2eq3dv 6721	. . . . . . . 8 |- ( r = R -> ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) = ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) )
19	13, 18	fveq12d 6197	. . . . . . 7 |- ( r = R -> ( ( N maDet r ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) = ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) )
20	19	mpt2eq3dv 6721	. . . . . 6 |- ( r = R -> ( i e. N , j e. N |-> ( ( N maDet r ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) = ( i e. N , j e. N |-> ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) ) )
21	12, 20	mpteq12dv 4733	. . . . 5 |- ( r = R -> ( m e. ( Base ` ( N Mat r ) ) |-> ( i e. N , j e. N |-> ( ( N maDet r ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) ) = ( m e. ( Base ` ( N Mat R ) ) |-> ( i e. N , j e. N |-> ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) ) ) )
22		df-madu 20440	. . . . 5 |- maAdju = ( n e. _V , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( i e. n , j e. n |-> ( ( n maDet r ) ` ( k e. n , l e. n |-> if ( k = j , if ( l = i , ( 1r ` r ) , ( 0g ` r ) ) , ( k m l ) ) ) ) ) ) )
23		fvex 6201	. . . . . 6 |- ( Base ` ( N Mat R ) ) e. _V
24	23	mptex 6486	. . . . 5 |- ( m e. ( Base ` ( N Mat R ) ) |-> ( i e. N , j e. N |-> ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) ) ) e. _V
25	10, 21, 22, 24	ovmpt2 6796	. . . 4 |- ( ( N e. _V /\ R e. _V ) -> ( N maAdju R ) = ( m e. ( Base ` ( N Mat R ) ) |-> ( i e. N , j e. N |-> ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) ) ) )
26		madufval.b	. . . . . 6 |- B = ( Base ` A )
27		madufval.a	. . . . . . 7 |- A = ( N Mat R )
28	27	fveq2i 6194	. . . . . 6 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
29	26, 28	eqtri 2644	. . . . 5 |- B = ( Base ` ( N Mat R ) )
30		madufval.d	. . . . . . . 8 |- D = ( N maDet R )
31		madufval.o	. . . . . . . . . . . 12 |- .1. = ( 1r ` R )
32	31	a1i 11	. . . . . . . . . . 11 |- ( ( k e. N /\ l e. N ) -> .1. = ( 1r ` R ) )
33		madufval.z	. . . . . . . . . . . 12 |- .0. = ( 0g ` R )
34	33	a1i 11	. . . . . . . . . . 11 |- ( ( k e. N /\ l e. N ) -> .0. = ( 0g ` R ) )
35	32, 34	ifeq12d 4106	. . . . . . . . . 10 |- ( ( k e. N /\ l e. N ) -> if ( l = i , .1. , .0. ) = if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) )
36	35	ifeq1d 4104	. . . . . . . . 9 |- ( ( k e. N /\ l e. N ) -> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) = if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) )
37	36	mpt2eq3ia 6720	. . . . . . . 8 |- ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) = ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) )
38	30, 37	fveq12i 6196	. . . . . . 7 |- ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) = ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) )
39	38	a1i 11	. . . . . 6 |- ( ( i e. N /\ j e. N ) -> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) = ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) )
40	39	mpt2eq3ia 6720	. . . . 5 |- ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) = ( i e. N , j e. N |-> ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) )
41	29, 40	mpteq12i 4742	. . . 4 |- ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) = ( m e. ( Base ` ( N Mat R ) ) |-> ( i e. N , j e. N |-> ( ( N maDet R ) ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , ( 1r ` R ) , ( 0g ` R ) ) , ( k m l ) ) ) ) ) )
42	25, 41	syl6eqr 2674	. . 3 |- ( ( N e. _V /\ R e. _V ) -> ( N maAdju R ) = ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) )
43	22	reldmmpt2 6771	. . . . 5 |- Rel dom maAdju
44	43	ovprc 6683	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( N maAdju R ) = (/) )
45		df-mat 20214	. . . . . . . . . . 11 |- Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
46	45	reldmmpt2 6771	. . . . . . . . . 10 |- Rel dom Mat
47	46	ovprc 6683	. . . . . . . . 9 |- ( -. ( N e. _V /\ R e. _V ) -> ( N Mat R ) = (/) )
48	27, 47	syl5eq 2668	. . . . . . . 8 |- ( -. ( N e. _V /\ R e. _V ) -> A = (/) )
49	48	fveq2d 6195	. . . . . . 7 |- ( -. ( N e. _V /\ R e. _V ) -> ( Base ` A ) = ( Base ` (/) ) )
50		base0 15912	. . . . . . 7 |- (/) = ( Base ` (/) )
51	49, 26, 50	3eqtr4g 2681	. . . . . 6 |- ( -. ( N e. _V /\ R e. _V ) -> B = (/) )
52	51	mpteq1d 4738	. . . . 5 |- ( -. ( N e. _V /\ R e. _V ) -> ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) = ( m e. (/) |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) )
53		mpt0 6021	. . . . 5 |- ( m e. (/) |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) = (/)
54	52, 53	syl6eq 2672	. . . 4 |- ( -. ( N e. _V /\ R e. _V ) -> ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) = (/) )
55	44, 54	eqtr4d 2659	. . 3 |- ( -. ( N e. _V /\ R e. _V ) -> ( N maAdju R ) = ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) ) )
56	42, 55	pm2.61i 176	. 2 |- ( N maAdju R ) = ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) )
57	1, 56	eqtri 2644	1 |- J = ( m e. B |-> ( i e. N , j e. N |-> ( D ` ( k e. N , l e. N |-> if ( k = j , if ( l = i , .1. , .0. ) , ( k m l ) ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   <.cop 4183   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   ndxcnx 15854   sSet csts 15855   Basecbs 15857   .rcmulr 15942   0gc0g 16100   1rcur 18501   freeLMod cfrlm 20090   maMul cmmul 20189   Mat cmat 20213   maDet cmdat 20390   maAdju cmadu 20438

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-madu 20440

This theorem is referenced by:  maduval  20444  maduf  20447