Theorem pmatcollpw3lem 19502

Description: Lemma for pmatcollpw3 19503 and pmatcollpw3fi 19504: 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 5213	. . . . . . . . 9 |- ( x = y -> dom x = dom y )
2	1	dmeqd 5215	. . . . . . . 8 |- ( x = y -> dom dom x = dom dom y )
3		oveq 6302	. . . . . . . . . 10 |- ( x = y -> ( i x j ) = ( i y j ) )
4	3	fveq2d 5876	. . . . . . . . 9 |- ( x = y -> (coe1 ` ( i x j ) ) = (coe1 ` ( i y j ) ) )
5	4	fveq1d 5874	. . . . . . . 8 |- ( x = y -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` k ) )
6	2, 2, 5	mpt2eq123dv 6358	. . . . . . 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 5872	. . . . . . . 8 |- ( k = l -> ( (coe1 ` ( i y j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` l ) )
8	7	mpt2eq3dv 6362	. . . . . . 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 6376	. . . . . 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 6730	. . . . . . . . . . 11 |- ( y e. B -> dom y e. _V )
11		dmexg 6730	. . . . . . . . . . 11 |- ( dom y e. _V -> dom dom y e. _V )
12	10, 11	syl 16	. . . . . . . . . 10 |- ( y e. B -> dom dom y e. _V )
13	12, 12	jca 532	. . . . . . . . 9 |- ( y e. B -> ( dom dom y e. _V /\ dom dom y e. _V ) )
14	13	ad2antrl 727	. . . . . . . 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 6875	. . . . . . . 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 16	. . . . . . 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 2878	. . . . . 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 757	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I =/= (/) )
19		nn0ex 10822	. . . . . . . 8 |- NN0 e. _V
20	19	ssex 4600	. . . . . . 7 |- ( I C_ NN0 -> I e. _V )
21	20	ad2antrl 727	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I e. _V )
22		simp3 998	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> M e. B )
23	22	adantr 465	. . . . . 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 5872	. . . . . . . . 9 |- ( l = k -> ( (coe1 ` ( i y j ) ) ` l ) = ( (coe1 ` ( i y j ) ) ` k ) )
26	25	mpt2eq3dv 6362	. . . . . . . 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 3843	. . . . . . 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 2466	. . . . . . . . 9 |- ( x = y <-> y = x )
29		eqcom 2466	. . . . . . . . 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 265	. . . . . . . 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 3436	. . . . . . 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 2514	. . . . . 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 4548	. . . . 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 2514	. . . 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 5213	. . . . . . . . . . 11 |- ( x = M -> dom x = dom M )
36	35	dmeqd 5215	. . . . . . . . . 10 |- ( x = M -> dom dom x = dom dom M )
37		oveq 6302	. . . . . . . . . . . 12 |- ( x = M -> ( i x j ) = ( i M j ) )
38	37	fveq2d 5876	. . . . . . . . . . 11 |- ( x = M -> (coe1 ` ( i x j ) ) = (coe1 ` ( i M j ) ) )
39	38	fveq1d 5874	. . . . . . . . . 10 |- ( x = M -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
40	36, 36, 39	mpt2eq123dv 6358	. . . . . . . . 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 466	. . . . . . . 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 3457	. . . . . . 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 2457	. . . . . . . . . . . . 13 |- ( Base ` P ) = ( Base ` P )
45		pmatcollpw.b	. . . . . . . . . . . . 13 |- B = ( Base ` C )
46	43, 44, 45	matbas2i 19142	. . . . . . . . . . . 12 |- ( M e. B -> M e. ( ( Base ` P ) ^m ( N X. N ) ) )
47		elmapi 7459	. . . . . . . . . . . 12 |- ( M e. ( ( Base ` P ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` P ) )
48		fdm 5741	. . . . . . . . . . . . . 14 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom M = ( N X. N ) )
49	48	dmeqd 5215	. . . . . . . . . . . . 13 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom dom M = dom ( N X. N ) )
50		dmxpid 5232	. . . . . . . . . . . . 13 |- dom ( N X. N ) = N
51	49, 50	syl6req 2515	. . . . . . . . . . . 12 |- ( M : ( N X. N ) --> ( Base ` P ) -> N = dom dom M )
52	46, 47, 51	3syl 20	. . . . . . . . . . 11 |- ( M e. B -> N = dom dom M )
53	52	3ad2ant3 1019	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> N = dom dom M )
54	53	adantr 465	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> N = dom dom M )
55		simpr 461	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> m = M )
56	55	oveqd 6313	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
57	56	fveq2d 5876	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
58	57	fveq1d 5874	. . . . . . . . 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 6358	. . . . . . . 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 3457	. . . . . . 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 2501	. . . . . 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 465	. . . . 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 4544	. . . 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 2498	. . 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 6302	. . . . . . . . . . . 12 |- ( m = M -> ( i m j ) = ( i M j ) )
66	65	adantl 466	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
67	66	fveq2d 5876	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
68	67	fveq1d 5874	. . . . . . . . 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 6362	. . . . . . . 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 3457	. . . . . . 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 725	. . . . . 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 2457	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
74		pmatcollpw3.d	. . . . . . 7 |- D = ( Base ` A )
75		simpll1 1035	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> N e. Fin )
76		simpll2 1036	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> R e. CRing )
77		simp2 997	. . . . . . . . 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 998	. . . . . . . . 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 465	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> M e. B )
80	79	3ad2ant1 1017	. . . . . . . . 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 19146	. . . . . . . 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 3493	. . . . . . . . . . 11 |- ( I C_ NN0 -> ( k e. I -> k e. NN0 ) )
83	82	ad2antrl 727	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I -> k e. NN0 ) )
84	83	imp 429	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> k e. NN0 )
85	84	3ad2ant1 1017	. . . . . . . 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 2457	. . . . . . . . 9 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
87		pmatcollpw.p	. . . . . . . . 9 |- P = (Poly1 ` R )
88	86, 44, 87, 73	coe1fvalcl 18469	. . . . . . . 8 |- ( ( ( i M j ) e. ( Base ` P ) /\ k e. NN0 ) -> ( (coe1 ` ( i M j ) ) ` k ) e. ( Base ` R ) )
89	81, 85, 88	syl2anc 661	. . . . . . 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 19143	. . . . . 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 2545	. . . . 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 2457	. . . . 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 6056	. . . 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 5882	. . . . . . 7 |- ( Base ` A ) e. _V
95	74, 94	eqeltri 2541	. . . . . 6 |- D e. _V
96	95	a1i 11	. . . . 5 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> D e. _V )
97	20	adantr 465	. . . . 5 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I e. _V )
98		elmapg 7451	. . . . 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 477	. . . 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 232	. . 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 2545	. 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 5871	. . . . . . . . . . 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 466	. . . . . . . . . 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 465	. . . . . . . . 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 2457	. . . . . . . . . . . 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 6730	. . . . . . . . . . . . . . . . 17 |- ( x e. B -> dom x e. _V )
107		dmexg 6730	. . . . . . . . . . . . . . . . 17 |- ( dom x e. _V -> dom dom x e. _V )
108	106, 107	syl 16	. . . . . . . . . . . . . . . 16 |- ( x e. B -> dom dom x e. _V )
109	108, 108	jca 532	. . . . . . . . . . . . . . 15 |- ( x e. B -> ( dom dom x e. _V /\ dom dom x e. _V ) )
110	109	ad2antrl 727	. . . . . . . . . . . . . 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 6875	. . . . . . . . . . . . . 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 16	. . . . . . . . . . . . 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 2878	. . . . . . . . . . . 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 465	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> I e. _V )
115	23	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> M e. B )
116		simpr 461	. . . . . . . . . . . 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 19482	. . . . . . . . . . . . . 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 5279	. . . . . . . . . . . . 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 3519	. . . . . . . . . . . . . . . . 17 |- B C_ _V
121	120	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> B C_ _V )
122		simpl 457	. . . . . . . . . . . . . . . 16 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I C_ NN0 )
123	121, 122	anim12i 566	. . . . . . . . . . . . . . 15 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( B C_ _V /\ I C_ NN0 ) )
124	123	adantr 465	. . . . . . . . . . . . . 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 6399	. . . . . . . . . . . . . 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 16	. . . . . . . . . . . . 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 2511	. . . . . . . . . . . 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 6313	. . . . . . . . . . 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 2498	. . . . . . . . . 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 714	. . . . . . . . 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 2498	. . . . . . . 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 5876	. . . . . . 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 725	. . . . . . . . 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 6441	. . . . . . . . 9 |- ( ( M e. B /\ n e. I ) -> ( M ( decompPMat |` ( B X. I ) ) n ) = ( M decompPMat n ) )
135	133, 134	sylan 471	. . . . . . . 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 5876	. . . . . . 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 2498	. . . . . 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 6312	. . . . 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 4543	. . . 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 6312	. . 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 2471	. 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 3214	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 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   E.wrex 2808   _Vcvv 3109   [_csb 3430   C_ wss 3471   (/)c0 3793   |-> cmpt 4515   X. cxp 5006   dom cdm 5008   |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298  curry ccur 7012   ^m cmap 7438   Fincfn 7535   NN0cn0 10816   Basecbs 14735   .scvsca 14807   gsum_g cgsu 14949  ._gcmg 16274  mulGrpcmgp 17359   CRingccrg 17417  var₁cv1 18433  Poly₁cpl1 18434  coe₁cco1 18435   Mat cmat 19127   matToPolyMat cmat2pmat 19423   decompPMat cdecpmat 19481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-cur 7014  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-sets 14741  df-ress 14742  df-plusg 14816  df-mulr 14817  df-sca 14819  df-vsca 14820  df-ip 14821  df-tset 14822  df-ple 14823  df-ds 14825  df-hom 14827  df-cco 14828  df-0g 14950  df-prds 14956  df-pws 14958  df-sra 18036  df-rgmod 18037  df-psr 18223  df-opsr 18227  df-psr1 18437  df-ply1 18439  df-coe1 18440  df-dsmm 18981  df-frlm 18996  df-mat 19128  df-decpmat 19482

This theorem is referenced by:  pmatcollpw3  19503  pmatcollpw3fi  19504