Theorem pmatcollpw3lem 20588

Description: Lemma for pmatcollpw3 20589 and pmatcollpw3fi 20590: Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 8-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw.p	|- P = (Poly1 ` R )
pmatcollpw.c	|- C = ( N Mat P )
pmatcollpw.b	|- B = ( Base ` C )
pmatcollpw.m	|- .* = ( .s ` C )
pmatcollpw.e	|- .^ = (.g ` (mulGrp ` P ) )
pmatcollpw.x	|- X = (var1 ` R )
pmatcollpw.t	|- T = ( N matToPolyMat R )
pmatcollpw3.a	|- A = ( N Mat R )
pmatcollpw3.d	|- D = ( Base ` A )

Assertion

Ref	Expression
pmatcollpw3lem	|- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( M = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) ) -> E. f e. ( D ^m I ) M = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) ) ) ) )

Distinct variable groups:   B, n   n, M   n, N   P, n   R, n   n, X   .^ , n   C, n   B, f   C, f, n   D, f   f, I, n   f, M   f, N   R, f   T, f   f, X   .^ , f   .* , f

Allowed substitution hints:   A( f, n)   D( n)   P( f)   T( n)   .* ( n)

Proof of Theorem pmatcollpw3lem

Dummy variables i j k l x y m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5324	. . . . . . . . 9 |- ( x = y -> dom x = dom y )
2	1	dmeqd 5326	. . . . . . . 8 |- ( x = y -> dom dom x = dom dom y )
3		oveq 6656	. . . . . . . . . 10 |- ( x = y -> ( i x j ) = ( i y j ) )
4	3	fveq2d 6195	. . . . . . . . 9 |- ( x = y -> (coe1 ` ( i x j ) ) = (coe1 ` ( i y j ) ) )
5	4	fveq1d 6193	. . . . . . . 8 |- ( x = y -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` k ) )
6	2, 2, 5	mpt2eq123dv 6717	. . . . . . 7 |- ( x = y -> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) )
7		fveq2 6191	. . . . . . . 8 |- ( k = l -> ( (coe1 ` ( i y j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` l ) )
8	7	mpt2eq3dv 6721	. . . . . . 7 |- ( k = l -> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) = ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) )
9	6, 8	cbvmpt2v 6735	. . . . . 6 |- ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) = ( y e. B , l e. I |-> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) )
10		dmexg 7097	. . . . . . . . . . 11 |- ( y e. B -> dom y e. _V )
11		dmexg 7097	. . . . . . . . . . 11 |- ( dom y e. _V -> dom dom y e. _V )
12	10, 11	syl 17	. . . . . . . . . 10 |- ( y e. B -> dom dom y e. _V )
13	12, 12	jca 554	. . . . . . . . 9 |- ( y e. B -> ( dom dom y e. _V /\ dom dom y e. _V ) )
14	13	ad2antrl 764	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ ( y e. B /\ l e. I ) ) -> ( dom dom y e. _V /\ dom dom y e. _V ) )
15		mpt2exga 7246	. . . . . . . 8 |- ( ( dom dom y e. _V /\ dom dom y e. _V ) -> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) e. _V )
16	14, 15	syl 17	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ ( y e. B /\ l e. I ) ) -> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) e. _V )
17	16	ralrimivva 2971	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> A. y e. B A. l e. I ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) e. _V )
18		simprr 796	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I =/= (/) )
19		nn0ex 11298	. . . . . . . 8 |- NN0 e. _V
20	19	ssex 4802	. . . . . . 7 |- ( I C_ NN0 -> I e. _V )
21	20	ad2antrl 764	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I e. _V )
22		simp3 1063	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> M e. B )
23	22	adantr 481	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> M e. B )
24	9, 17, 18, 21, 23	mpt2curryvald 7396	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) = ( l e. I |-> [_ M / y ]_ ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) ) )
25		fveq2 6191	. . . . . . . . 9 |- ( l = k -> ( (coe1 ` ( i y j ) ) ` l ) = ( (coe1 ` ( i y j ) ) ` k ) )
26	25	mpt2eq3dv 6721	. . . . . . . 8 |- ( l = k -> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) = ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) )
27	26	csbeq2dv 3992	. . . . . . 7 |- ( l = k -> [_ M / y ]_ ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) = [_ M / y ]_ ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) )
28		eqcom 2629	. . . . . . . . 9 |- ( x = y <-> y = x )
29		eqcom 2629	. . . . . . . . 9 |- ( ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) <-> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) = ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) )
30	6, 28, 29	3imtr3i 280	. . . . . . . 8 |- ( y = x -> ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) = ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) )
31	30	cbvcsbv 3539	. . . . . . 7 |- [_ M / y ]_ ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` k ) ) = [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) )
32	27, 31	syl6eq 2672	. . . . . 6 |- ( l = k -> [_ M / y ]_ ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) = [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) )
33	32	cbvmptv 4750	. . . . 5 |- ( l e. I |-> [_ M / y ]_ ( i e. dom dom y , j e. dom dom y |-> ( (coe1 ` ( i y j ) ) ` l ) ) ) = ( k e. I |-> [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) )
34	24, 33	syl6eq 2672	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) = ( k e. I |-> [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) )
35		dmeq 5324	. . . . . . . . . . 11 |- ( x = M -> dom x = dom M )
36	35	dmeqd 5326	. . . . . . . . . 10 |- ( x = M -> dom dom x = dom dom M )
37		oveq 6656	. . . . . . . . . . . 12 |- ( x = M -> ( i x j ) = ( i M j ) )
38	37	fveq2d 6195	. . . . . . . . . . 11 |- ( x = M -> (coe1 ` ( i x j ) ) = (coe1 ` ( i M j ) ) )
39	38	fveq1d 6193	. . . . . . . . . 10 |- ( x = M -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
40	36, 36, 39	mpt2eq123dv 6717	. . . . . . . . 9 |- ( x = M -> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
41	40	adantl 482	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ x = M ) -> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
42	22, 41	csbied 3560	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
43		pmatcollpw.c	. . . . . . . . . . . . 13 |- C = ( N Mat P )
44		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` P ) = ( Base ` P )
45		pmatcollpw.b	. . . . . . . . . . . . 13 |- B = ( Base ` C )
46	43, 44, 45	matbas2i 20228	. . . . . . . . . . . 12 |- ( M e. B -> M e. ( ( Base ` P ) ^m ( N X. N ) ) )
47		elmapi 7879	. . . . . . . . . . . 12 |- ( M e. ( ( Base ` P ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` P ) )
48		fdm 6051	. . . . . . . . . . . . . 14 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom M = ( N X. N ) )
49	48	dmeqd 5326	. . . . . . . . . . . . 13 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom dom M = dom ( N X. N ) )
50		dmxpid 5345	. . . . . . . . . . . . 13 |- dom ( N X. N ) = N
51	49, 50	syl6req 2673	. . . . . . . . . . . 12 |- ( M : ( N X. N ) --> ( Base ` P ) -> N = dom dom M )
52	46, 47, 51	3syl 18	. . . . . . . . . . 11 |- ( M e. B -> N = dom dom M )
53	52	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> N = dom dom M )
54	53	adantr 481	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> N = dom dom M )
55		simpr 477	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> m = M )
56	55	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
57	56	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
58	57	fveq1d 6193	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( (coe1 ` ( i m j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
59	54, 54, 58	mpt2eq123dv 6717	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
60	22, 59	csbied 3560	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. dom dom M , j e. dom dom M |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
61	42, 60	eqtr4d 2659	. . . . . 6 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
62	61	adantr 481	. . . . 5 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) = [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
63	62	mpteq2dv 4745	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I |-> [_ M / x ]_ ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) = ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) )
64	34, 63	eqtrd 2656	. . 3 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) = ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) )
65		oveq 6656	. . . . . . . . . . . 12 |- ( m = M -> ( i m j ) = ( i M j ) )
66	65	adantl 482	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
67	66	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
68	67	fveq1d 6193	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( (coe1 ` ( i m j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
69	68	mpt2eq3dv 6721	. . . . . . . 8 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
70	22, 69	csbied 3560	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
71	70	ad2antrr 762	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) = ( i e. N , j e. N |-> ( (coe1 ` ( i M j ) ) ` k ) ) )
72		pmatcollpw3.a	. . . . . . 7 |- A = ( N Mat R )
73		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
74		pmatcollpw3.d	. . . . . . 7 |- D = ( Base ` A )
75		simpll1 1100	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> N e. Fin )
76		simpll2 1101	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> R e. CRing )
77		simp2 1062	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) /\ i e. N /\ j e. N ) -> i e. N )
78		simp3 1063	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) /\ i e. N /\ j e. N ) -> j e. N )
79	23	adantr 481	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> M e. B )
80	79	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) /\ i e. N /\ j e. N ) -> M e. B )
81	43, 44, 45, 77, 78, 80	matecld 20232	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) /\ i e. N /\ j e. N ) -> ( i M j ) e. ( Base ` P ) )
82		ssel 3597	. . . . . . . . . . 11 |- ( I C_ NN0 -> ( k e. I -> k e. NN0 ) )
83	82	ad2antrl 764	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I -> k e. NN0 ) )
84	83	imp 445	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> k e. NN0 )
85	84	3ad2ant1 1082	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) /\ i e. N /\ j e. N ) -> k e. NN0 )
86		eqid 2622	. . . . . . . . 9 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
87		pmatcollpw.p	. . . . . . . . 9 |- P = (Poly1 ` R )
88	86, 44, 87, 73	coe1fvalcl 19582	. . . . . . . 8 |- ( ( ( i M j ) e. ( Base ` P ) /\ k e. NN0 ) -> ( (coe1 ` ( i M j ) ) ` k ) e. ( Base ` R ) )
89	81, 85, 88	syl2anc 693	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) /\ i e. N /\ j e. N ) -> ( (coe1 ` ( i M j ) ) ` k ) e. ( Base ` R ) )
90	72, 73, 74, 75, 76, 89	matbas2d 20229	. . . . . 6 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> ( i e. N , j e. N |-> ( (coe1 ` ( i M j ) ) ` k ) ) e. D )
91	71, 90	eqeltrd 2701	. . . . 5 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) e. D )
92		eqid 2622	. . . . 5 |- ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) = ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) )
93	91, 92	fmptd 6385	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) : I --> D )
94		fvex 6201	. . . . . . 7 |- ( Base ` A ) e. _V
95	74, 94	eqeltri 2697	. . . . . 6 |- D e. _V
96	95	a1i 11	. . . . 5 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> D e. _V )
97	20	adantr 481	. . . . 5 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I e. _V )
98		elmapg 7870	. . . . 5 |- ( ( D e. _V /\ I e. _V ) -> ( ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) e. ( D ^m I ) <-> ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) : I --> D ) )
99	96, 97, 98	syl2an 494	. . . 4 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) e. ( D ^m I ) <-> ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) : I --> D ) )
100	93, 99	mpbird 247	. . 3 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I |-> [_ M / m ]_ ( i e. N , j e. N |-> ( (coe1 ` ( i m j ) ) ` k ) ) ) e. ( D ^m I ) )
101	64, 100	eqeltrd 2701	. 2 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) e. ( D ^m I ) )
102		fveq1 6190	. . . . . . . . . . 11 |- ( f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) -> ( f ` n ) = ( (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ` n ) )
103	102	adantl 482	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) -> ( f ` n ) = ( (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ` n ) )
104	103	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( f ` n ) = ( (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ` n ) )
105		eqid 2622	. . . . . . . . . . . 12 |- ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) = ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) )
106		dmexg 7097	. . . . . . . . . . . . . . . . 17 |- ( x e. B -> dom x e. _V )
107		dmexg 7097	. . . . . . . . . . . . . . . . 17 |- ( dom x e. _V -> dom dom x e. _V )
108	106, 107	syl 17	. . . . . . . . . . . . . . . 16 |- ( x e. B -> dom dom x e. _V )
109	108, 108	jca 554	. . . . . . . . . . . . . . 15 |- ( x e. B -> ( dom dom x e. _V /\ dom dom x e. _V ) )
110	109	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) /\ ( x e. B /\ k e. I ) ) -> ( dom dom x e. _V /\ dom dom x e. _V ) )
111		mpt2exga 7246	. . . . . . . . . . . . . 14 |- ( ( dom dom x e. _V /\ dom dom x e. _V ) -> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) e. _V )
112	110, 111	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) /\ ( x e. B /\ k e. I ) ) -> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) e. _V )
113	112	ralrimivva 2971	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> A. x e. B A. k e. I ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) e. _V )
114	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> I e. _V )
115	23	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> M e. B )
116		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> n e. I )
117	105, 113, 114, 115, 116	fvmpt2curryd 7397	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> ( (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ` n ) = ( M ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) n ) )
118		df-decpmat 20568	. . . . . . . . . . . . . 14 |- decompPMat = ( x e. _V , k e. NN0 |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) )
119	118	reseq1i 5392	. . . . . . . . . . . . 13 |- ( decompPMat |` ( B X. I ) ) = ( ( x e. _V , k e. NN0 |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) |` ( B X. I ) )
120		ssv 3625	. . . . . . . . . . . . . . . . 17 |- B C_ _V
121	120	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> B C_ _V )
122		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I C_ NN0 )
123	121, 122	anim12i 590	. . . . . . . . . . . . . . 15 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( B C_ _V /\ I C_ NN0 ) )
124	123	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> ( B C_ _V /\ I C_ NN0 ) )
125		resmpt2 6758	. . . . . . . . . . . . . 14 |- ( ( B C_ _V /\ I C_ NN0 ) -> ( ( x e. _V , k e. NN0 |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) |` ( B X. I ) ) = ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) )
126	124, 125	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> ( ( x e. _V , k e. NN0 |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) |` ( B X. I ) ) = ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) )
127	119, 126	syl5req 2669	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) = ( decompPMat |` ( B X. I ) ) )
128	127	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> ( M ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) n ) = ( M ( decompPMat |` ( B X. I ) ) n ) )
129	117, 128	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> ( (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ` n ) = ( M ( decompPMat |` ( B X. I ) ) n ) )
130	129	adantlr 751	. . . . . . . . 9 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ` n ) = ( M ( decompPMat |` ( B X. I ) ) n ) )
131	104, 130	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( f ` n ) = ( M ( decompPMat |` ( B X. I ) ) n ) )
132	131	fveq2d 6195	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( T ` ( f ` n ) ) = ( T ` ( M ( decompPMat |` ( B X. I ) ) n ) ) )
133	22	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) -> M e. B )
134		ovres 6800	. . . . . . . . 9 |- ( ( M e. B /\ n e. I ) -> ( M ( decompPMat |` ( B X. I ) ) n ) = ( M decompPMat n ) )
135	133, 134	sylan 488	. . . . . . . 8 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( M ( decompPMat |` ( B X. I ) ) n ) = ( M decompPMat n ) )
136	135	fveq2d 6195	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( T ` ( M ( decompPMat |` ( B X. I ) ) n ) ) = ( T ` ( M decompPMat n ) ) )
137	132, 136	eqtrd 2656	. . . . . 6 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( T ` ( f ` n ) ) = ( T ` ( M decompPMat n ) ) )
138	137	oveq2d 6666	. . . . 5 |- ( ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) /\ n e. I ) -> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) = ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) )
139	138	mpteq2dva 4744	. . . 4 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) -> ( n e. I |-> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) ) = ( n e. I |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) )
140	139	oveq2d 6666	. . 3 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) -> ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) ) ) = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) ) )
141	140	eqeq2d 2632	. 2 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ f = (curry ( x e. B , k e. I |-> ( i e. dom dom x , j e. dom dom x |-> ( (coe1 ` ( i x j ) ) ` k ) ) ) ` M ) ) -> ( M = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) ) ) <-> M = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) ) ) )
142	101, 141	rspcedv 3313	1 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( M = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( M decompPMat n ) ) ) ) ) -> E. f e. ( D ^m I ) M = ( C gsumg ( n e. I |-> ( ( n .^ X ) .* ( T ` ( f ` n ) ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   [_csb 3533   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  curry ccur 7391   ^m cmap 7857   Fincfn 7955   NN0cn0 11292   Basecbs 15857   .scvsca 15945   gsum_g cgsu 16101  ._gcmg 17540  mulGrpcmgp 18489   CRingccrg 18548  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   matToPolyMat cmat2pmat 20509   decompPMat cdecpmat 20567

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  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-cur 7393  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-psr 19356  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-decpmat 20568

This theorem is referenced by:  pmatcollpw3  20589  pmatcollpw3fi  20590