Theorem pmatcollpw3lem 19884

Description: Lemma for pmatcollpw3 19885 and pmatcollpw3fi 19886: 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 5040	. . . . . . . . 9 |- ( x = y -> dom x = dom y )
2	1	dmeqd 5042	. . . . . . . 8 |- ( x = y -> dom dom x = dom dom y )
3		oveq 6314	. . . . . . . . . 10 |- ( x = y -> ( i x j ) = ( i y j ) )
4	3	fveq2d 5883	. . . . . . . . 9 |- ( x = y -> (coe1 ` ( i x j ) ) = (coe1 ` ( i y j ) ) )
5	4	fveq1d 5881	. . . . . . . 8 |- ( x = y -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` k ) )
6	2, 2, 5	mpt2eq123dv 6372	. . . . . . 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 5879	. . . . . . . 8 |- ( k = l -> ( (coe1 ` ( i y j ) ) ` k ) = ( (coe1 ` ( i y j ) ) ` l ) )
8	7	mpt2eq3dv 6376	. . . . . . 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 6390	. . . . . 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 6743	. . . . . . . . . . 11 |- ( y e. B -> dom y e. _V )
11		dmexg 6743	. . . . . . . . . . 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 541	. . . . . . . . 9 |- ( y e. B -> ( dom dom y e. _V /\ dom dom y e. _V ) )
14	13	ad2antrl 742	. . . . . . . 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 6888	. . . . . . . 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 2814	. . . . . 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 774	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I =/= (/) )
19		nn0ex 10899	. . . . . . . 8 |- NN0 e. _V
20	19	ssex 4540	. . . . . . 7 |- ( I C_ NN0 -> I e. _V )
21	20	ad2antrl 742	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> I e. _V )
22		simp3 1032	. . . . . . 7 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> M e. B )
23	22	adantr 472	. . . . . 6 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> M e. B )
24	9, 17, 18, 21, 23	mpt2curryvald 7035	. . . . 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 5879	. . . . . . . . 9 |- ( l = k -> ( (coe1 ` ( i y j ) ) ` l ) = ( (coe1 ` ( i y j ) ) ` k ) )
26	25	mpt2eq3dv 6376	. . . . . . . 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 3785	. . . . . . 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 2478	. . . . . . . . 9 |- ( x = y <-> y = x )
29		eqcom 2478	. . . . . . . . 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 273	. . . . . . . 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 3355	. . . . . . 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 2521	. . . . . 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 4488	. . . . 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 2521	. . . 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 5040	. . . . . . . . . . 11 |- ( x = M -> dom x = dom M )
36	35	dmeqd 5042	. . . . . . . . . 10 |- ( x = M -> dom dom x = dom dom M )
37		oveq 6314	. . . . . . . . . . . 12 |- ( x = M -> ( i x j ) = ( i M j ) )
38	37	fveq2d 5883	. . . . . . . . . . 11 |- ( x = M -> (coe1 ` ( i x j ) ) = (coe1 ` ( i M j ) ) )
39	38	fveq1d 5881	. . . . . . . . . 10 |- ( x = M -> ( (coe1 ` ( i x j ) ) ` k ) = ( (coe1 ` ( i M j ) ) ` k ) )
40	36, 36, 39	mpt2eq123dv 6372	. . . . . . . . 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 473	. . . . . . . 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 3376	. . . . . . 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 2471	. . . . . . . . . . . . 13 |- ( Base ` P ) = ( Base ` P )
45		pmatcollpw.b	. . . . . . . . . . . . 13 |- B = ( Base ` C )
46	43, 44, 45	matbas2i 19524	. . . . . . . . . . . 12 |- ( M e. B -> M e. ( ( Base ` P ) ^m ( N X. N ) ) )
47		elmapi 7511	. . . . . . . . . . . 12 |- ( M e. ( ( Base ` P ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` P ) )
48		fdm 5745	. . . . . . . . . . . . . 14 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom M = ( N X. N ) )
49	48	dmeqd 5042	. . . . . . . . . . . . 13 |- ( M : ( N X. N ) --> ( Base ` P ) -> dom dom M = dom ( N X. N ) )
50		dmxpid 5060	. . . . . . . . . . . . 13 |- dom ( N X. N ) = N
51	49, 50	syl6req 2522	. . . . . . . . . . . 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 1053	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> N = dom dom M )
54	53	adantr 472	. . . . . . . . 9 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> N = dom dom M )
55		simpr 468	. . . . . . . . . . . 12 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> m = M )
56	55	oveqd 6325	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
57	56	fveq2d 5883	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
58	57	fveq1d 5881	. . . . . . . . 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 6372	. . . . . . . 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 3376	. . . . . . 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 2508	. . . . . 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 472	. . . . 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 4483	. . . 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 2505	. . 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 6314	. . . . . . . . . . . 12 |- ( m = M -> ( i m j ) = ( i M j ) )
66	65	adantl 473	. . . . . . . . . . 11 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> ( i m j ) = ( i M j ) )
67	66	fveq2d 5883	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ m = M ) -> (coe1 ` ( i m j ) ) = (coe1 ` ( i M j ) ) )
68	67	fveq1d 5881	. . . . . . . . 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 6376	. . . . . . . 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 3376	. . . . . . 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 740	. . . . . 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 2471	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
74		pmatcollpw3.d	. . . . . . 7 |- D = ( Base ` A )
75		simpll1 1069	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> N e. Fin )
76		simpll2 1070	. . . . . . 7 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> R e. CRing )
77		simp2 1031	. . . . . . . . 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 1032	. . . . . . . . 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 472	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> M e. B )
80	79	3ad2ant1 1051	. . . . . . . . 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 19528	. . . . . . . 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 3412	. . . . . . . . . . 11 |- ( I C_ NN0 -> ( k e. I -> k e. NN0 ) )
83	82	ad2antrl 742	. . . . . . . . . 10 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( k e. I -> k e. NN0 ) )
84	83	imp 436	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ k e. I ) -> k e. NN0 )
85	84	3ad2ant1 1051	. . . . . . . 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 2471	. . . . . . . . 9 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
87		pmatcollpw.p	. . . . . . . . 9 |- P = (Poly1 ` R )
88	86, 44, 87, 73	coe1fvalcl 18882	. . . . . . . 8 |- ( ( ( i M j ) e. ( Base ` P ) /\ k e. NN0 ) -> ( (coe1 ` ( i M j ) ) ` k ) e. ( Base ` R ) )
89	81, 85, 88	syl2anc 673	. . . . . . 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 19525	. . . . . 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 2549	. . . . 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 2471	. . . . 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 6061	. . . 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 5889	. . . . . . 7 |- ( Base ` A ) e. _V
95	74, 94	eqeltri 2545	. . . . . 6 |- D e. _V
96	95	a1i 11	. . . . 5 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> D e. _V )
97	20	adantr 472	. . . . 5 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I e. _V )
98		elmapg 7503	. . . . 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 485	. . . 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 240	. . 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 2549	. 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 5878	. . . . . . . . . . 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 473	. . . . . . . . . 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 472	. . . . . . . . 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 2471	. . . . . . . . . . . 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 6743	. . . . . . . . . . . . . . . . 17 |- ( x e. B -> dom x e. _V )
107		dmexg 6743	. . . . . . . . . . . . . . . . 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 541	. . . . . . . . . . . . . . 15 |- ( x e. B -> ( dom dom x e. _V /\ dom dom x e. _V ) )
110	109	ad2antrl 742	. . . . . . . . . . . . . 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 6888	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . 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 472	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> I e. _V )
115	23	adantr 472	. . . . . . . . . . . 12 |- ( ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) /\ n e. I ) -> M e. B )
116		simpr 468	. . . . . . . . . . . 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 7036	. . . . . . . . . . 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 19864	. . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . 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 3438	. . . . . . . . . . . . . . . . 17 |- B C_ _V
121	120	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> B C_ _V )
122		simpl 464	. . . . . . . . . . . . . . . 16 |- ( ( I C_ NN0 /\ I =/= (/) ) -> I C_ NN0 )
123	121, 122	anim12i 576	. . . . . . . . . . . . . . 15 |- ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ ( I C_ NN0 /\ I =/= (/) ) ) -> ( B C_ _V /\ I C_ NN0 ) )
124	123	adantr 472	. . . . . . . . . . . . . 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 6413	. . . . . . . . . . . . . 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 2518	. . . . . . . . . . . 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 6325	. . . . . . . . . . 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 2505	. . . . . . . . . 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 729	. . . . . . . . 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 2505	. . . . . . . 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 5883	. . . . . . 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 740	. . . . . . . . 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 6455	. . . . . . . . 9 |- ( ( M e. B /\ n e. I ) -> ( M ( decompPMat |` ( B X. I ) ) n ) = ( M decompPMat n ) )
135	133, 134	sylan 479	. . . . . . . 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 5883	. . . . . . 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 2505	. . . . . 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 6324	. . . . 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 4482	. . . 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 6324	. . 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 2481	. 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 3140	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   _Vcvv 3031   [_csb 3349   C_ wss 3390   (/)c0 3722   |-> cmpt 4454   X. cxp 4837   dom cdm 4839   |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310  curry ccur 7030   ^m cmap 7490   Fincfn 7587   NN0cn0 10893   Basecbs 15199   .scvsca 15272   gsum_g cgsu 15417  ._gcmg 16750  mulGrpcmgp 17801   CRingccrg 17859  var₁cv1 18846  Poly₁cpl1 18847  coe₁cco1 18848   Mat cmat 19509   matToPolyMat cmat2pmat 19805   decompPMat cdecpmat 19863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-cur 7032  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-0g 15418  df-prds 15424  df-pws 15426  df-sra 18473  df-rgmod 18474  df-psr 18657  df-opsr 18661  df-psr1 18850  df-ply1 18852  df-coe1 18853  df-dsmm 19372  df-frlm 19387  df-mat 19510  df-decpmat 19864

This theorem is referenced by:  pmatcollpw3  19885  pmatcollpw3fi  19886