Theorem pmatcollpw3lem 19807

Description: Lemma for pmatcollpw3 19808 and pmatcollpw3fi 19809: Write a polynomial matrix (over a commutative ring) as 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 5035	. . . . . . . . 9 |- ( x = y -> dom x = dom y )
2	1	dmeqd 5037	. . . . . . . 8 |- ( x = y -> dom dom x = dom dom y )
3		oveq 6296	. . . . . . . . . 10 |- ( x = y -> ( i x j ) = ( i y j ) )
4	3	fveq2d 5869	. . . . . . . . 9 |- ( x = y -> (coe1 ` ( i x j ) ) = (coe1 ` ( i y j ) ) )
5	4	fveq1d 5867	. . . . . . . 8 |- ( x = y -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` k ) )
6	2, 2, 5	mpt2eq123dv 6353	. . . . . . 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 5865	. . . . . . . 8 |- ( k = l -> ( (coe1 ` ( i y j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` l ) )
8	7	mpt2eq3dv 6357	. . . . . . 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 6371	. . . . . 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 6724	. . . . . . . . . . 11 |- ( y e. B -> dom y e. _V )
11		dmexg 6724	. . . . . . . . . . 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 535	. . . . . . . . 9 |- ( y e. B -> ( dom dom y e. _V /\ dom dom y e. _V ) )
14	13	ad2antrl 734	. . . . . . . 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 6869	. . . . . . . 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 2809	. . . . . 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 766	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I =/= (/) )
19		nn0ex 10875	. . . . . . . 8 |- NN0 e. _V
20	19	ssex 4547	. . . . . . 7 |- ( I C_ NN0 -> I e. _V )
21	20	ad2antrl 734	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I e. _V )
22		simp3 1010	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> M e. B )
23	22	adantr 467	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> M e. B )
24	9, 17, 18, 21, 23	mpt2curryvald 7017	. . . . 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 5865	. . . . . . . . 9 |- ( l = k -> ( (coe1 ` ( i y j ) ) ` l ) = ( (coe1 ` ( i y j ) ) ` k ) )
26	25	mpt2eq3dv 6357	. . . . . . . 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 3781	. . . . . . 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 2458	. . . . . . . . 9 |- ( x = y <-> y = x )
29		eqcom 2458	. . . . . . . . 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 269	. . . . . . . 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 3369	. . . . . . 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 2501	. . . . . 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 4495	. . . . 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 2501	. . . 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 5035	. . . . . . . . . . 11 |- ( x = M -> dom x = dom M )
36	35	dmeqd 5037	. . . . . . . . . 10 |- ( x = M -> dom dom x = dom dom M )
37		oveq 6296	. . . . . . . . . . . 12 |- ( x = M -> ( i x j ) = ( i M j ) )
38	37	fveq2d 5869	. . . . . . . . . . 11 |- ( x = M -> (coe1 ` ( i x j ) ) = (coe1 ` ( i M j ) ) )
39	38	fveq1d 5867	. . . . . . . . . 10 |- ( x = M -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
40	36, 36, 39	mpt2eq123dv 6353	. . . . . . . . 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 468	. . . . . . . 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 3390	. . . . . . 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 2451	. . . . . . . . . . . . 13 |- ( Base ` P ) = ( Base ` P )
45		pmatcollpw.b	. . . . . . . . . . . . 13 |- B = ( Base ` C )
46	43, 44, 45	matbas2i 19447	. . . . . . . . . . . 12 |- ( M e. B -> M e. ( ( Base ` P ) ^m ( N X. N ) ) )
47		elmapi 7493	. . . . . . . . . . . 12 |- ( M e. ( ( Base ` P ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` P ) )
48		fdm 5733	. . . . . . . . . . . . . 14 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom M = ( N X. N ) )
49	48	dmeqd 5037	. . . . . . . . . . . . 13 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom dom M = dom ( N X. N ) )
50		dmxpid 5054	. . . . . . . . . . . . 13 |- dom ( N X. N ) = N
51	49, 50	syl6req 2502	. . . . . . . . . . . 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 1031	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> N = dom dom M )
54	53	adantr 467	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> N = dom dom M )
55		simpr 463	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> m = M )
56	55	oveqd 6307	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
57	56	fveq2d 5869	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
58	57	fveq1d 5867	. . . . . . . . 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 6353	. . . . . . . 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 3390	. . . . . . 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 2488	. . . . . 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 467	. . . . 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 4490	. . . 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 2485	. . 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 6296	. . . . . . . . . . . 12 |- ( m = M -> ( i m j ) = ( i M j ) )
66	65	adantl 468	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
67	66	fveq2d 5869	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
68	67	fveq1d 5867	. . . . . . . . 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 6357	. . . . . . . 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 3390	. . . . . . 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 732	. . . . . 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 2451	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
74		pmatcollpw3.d	. . . . . . 7 |- D = ( Base ` A )
75		simpll1 1047	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> N e. Fin )
76		simpll2 1048	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> R e. CRing )
77		simp2 1009	. . . . . . . . 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 1010	. . . . . . . . 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 467	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> M e. B )
80	79	3ad2ant1 1029	. . . . . . . . 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 19451	. . . . . . . 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 3426	. . . . . . . . . . 11 |- ( I C_ NN0 -> ( k e. I -> k e. NN0 ) )
83	82	ad2antrl 734	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I -> k e. NN0 ) )
84	83	imp 431	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> k e. NN0 )
85	84	3ad2ant1 1029	. . . . . . . 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 2451	. . . . . . . . 9 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
87		pmatcollpw.p	. . . . . . . . 9 |- P = (Poly1 ` R )
88	86, 44, 87, 73	coe1fvalcl 18805	. . . . . . . 8 |- ( ( ( i M j ) e. ( Base ` P ) /\ k e. NN0 ) -> ( (coe1 ` ( i M j ) ) ` k ) e. ( Base ` R ) )
89	81, 85, 88	syl2anc 667	. . . . . . 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 19448	. . . . . 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 2529	. . . . 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 2451	. . . . 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 6046	. . . 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 5875	. . . . . . 7 |- ( Base ` A ) e. _V
95	74, 94	eqeltri 2525	. . . . . 6 |- D e. _V
96	95	a1i 11	. . . . 5 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> D e. _V )
97	20	adantr 467	. . . . 5 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I e. _V )
98		elmapg 7485	. . . . 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 480	. . . 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 236	. . 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 2529	. 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 5864	. . . . . . . . . . 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 468	. . . . . . . . . 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 467	. . . . . . . . 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 2451	. . . . . . . . . . . 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 6724	. . . . . . . . . . . . . . . . 17 |- ( x e. B -> dom x e. _V )
107		dmexg 6724	. . . . . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . 15 |- ( x e. B -> ( dom dom x e. _V /\ dom dom x e. _V ) )
110	109	ad2antrl 734	. . . . . . . . . . . . . 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 6869	. . . . . . . . . . . . . 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 2809	. . . . . . . . . . . 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 467	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> I e. _V )
115	23	adantr 467	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> M e. B )
116		simpr 463	. . . . . . . . . . . 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 7018	. . . . . . . . . . 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 19787	. . . . . . . . . . . . . 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 5101	. . . . . . . . . . . . 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 3452	. . . . . . . . . . . . . . . . 17 |- B C_ _V
121	120	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> B C_ _V )
122		simpl 459	. . . . . . . . . . . . . . . 16 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I C_ NN0 )
123	121, 122	anim12i 570	. . . . . . . . . . . . . . 15 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( B C_ _V /\ I C_ NN0 ) )
124	123	adantr 467	. . . . . . . . . . . . . 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 6394	. . . . . . . . . . . . . 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 2498	. . . . . . . . . . . 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 6307	. . . . . . . . . . 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 2485	. . . . . . . . . 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 721	. . . . . . . . 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 2485	. . . . . . . 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 5869	. . . . . . 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 732	. . . . . . . . 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 6436	. . . . . . . . 9 |- ( ( M e. B /\ n e. I ) -> ( M ( decompPMat |` ( B X. I ) ) n ) = ( M decompPMat n ) )
135	133, 134	sylan 474	. . . . . . . 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 5869	. . . . . . 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 2485	. . . . . 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 6306	. . . . 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 4489	. . . 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 6306	. . 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 2461	. 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 3154	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   _Vcvv 3045   [_csb 3363   C_ wss 3404   (/)c0 3731   |-> cmpt 4461   X. cxp 4832   dom cdm 4834   |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292  curry ccur 7012   ^m cmap 7472   Fincfn 7569   NN0cn0 10869   Basecbs 15121   .scvsca 15194   gsum_g cgsu 15339  ._gcmg 16672  mulGrpcmgp 17723   CRingccrg 17781  var₁cv1 18769  Poly₁cpl1 18770  coe₁cco1 18771   Mat cmat 19432   matToPolyMat cmat2pmat 19728   decompPMat cdecpmat 19786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-cur 7014  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-hom 15214  df-cco 15215  df-0g 15340  df-prds 15346  df-pws 15348  df-sra 18395  df-rgmod 18396  df-psr 18580  df-opsr 18584  df-psr1 18773  df-ply1 18775  df-coe1 18776  df-dsmm 19295  df-frlm 19310  df-mat 19433  df-decpmat 19787

This theorem is referenced by:  pmatcollpw3  19808  pmatcollpw3fi  19809