Theorem pmatcoe1fsupp 20506

Description: For a polynomial matrix there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)

Hypotheses

Ref	Expression
pmatcoe1fsupp.p	|- P = (Poly1 ` R )
pmatcoe1fsupp.c	|- C = ( N Mat P )
pmatcoe1fsupp.b	|- B = ( Base ` C )
pmatcoe1fsupp.0	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
pmatcoe1fsupp	|- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> E. s e. NN0 A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = .0. ) )

Distinct variable groups:   B, i, j, s, x   i, M, j, s, x   i, N, j, s, x   R, i, j, s, x   .0. , i, j, s, x

Allowed substitution hints:   C( x, i, j, s)   P( x, i, j, s)

Proof of Theorem pmatcoe1fsupp

Dummy variables v u w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . . . 6 |- { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } C_ ( ( Base ` R ) ^m NN0 )
2	1	a1i 11	. . . . 5 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } C_ ( ( Base ` R ) ^m NN0 ) )
3	2	olcd 408	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } C_ ( ( Base ` R ) ^m NN0 ) \/ { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } C_ ( ( Base ` R ) ^m NN0 ) ) )
4		inss 3842	. . . 4 |- ( ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } C_ ( ( Base ` R ) ^m NN0 ) \/ { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } C_ ( ( Base ` R ) ^m NN0 ) ) -> ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) C_ ( ( Base ` R ) ^m NN0 ) )
5	3, 4	syl 17	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) C_ ( ( Base ` R ) ^m NN0 ) )
6		xpfi 8231	. . . . . . 7 |- ( ( N e. Fin /\ N e. Fin ) -> ( N X. N ) e. Fin )
7	6	anidms 677	. . . . . 6 |- ( N e. Fin -> ( N X. N ) e. Fin )
8		snfi 8038	. . . . . . . 8 |- { (coe1 ` ( M ` u ) ) } e. Fin
9	8	a1i 11	. . . . . . 7 |- ( ( N e. Fin /\ u e. ( N X. N ) ) -> { (coe1 ` ( M ` u ) ) } e. Fin )
10	9	ralrimiva 2966	. . . . . 6 |- ( N e. Fin -> A. u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } e. Fin )
11	7, 10	jca 554	. . . . 5 |- ( N e. Fin -> ( ( N X. N ) e. Fin /\ A. u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } e. Fin ) )
12	11	3ad2ant1 1082	. . . 4 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( ( N X. N ) e. Fin /\ A. u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } e. Fin ) )
13		iunfi 8254	. . . 4 |- ( ( ( N X. N ) e. Fin /\ A. u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } e. Fin ) -> U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } e. Fin )
14		infi 8184	. . . 4 |- ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } e. Fin -> ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) e. Fin )
15	12, 13, 14	3syl 18	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) e. Fin )
16		pmatcoe1fsupp.0	. . . . 5 |- .0. = ( 0g ` R )
17		fvex 6201	. . . . 5 |- ( 0g ` R ) e. _V
18	16, 17	eqeltri 2697	. . . 4 |- .0. e. _V
19	18	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> .0. e. _V )
20		elin 3796	. . . . . 6 |- ( w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) <-> ( w e. U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } /\ w e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) )
21		breq1 4656	. . . . . . . 8 |- ( v = w -> ( v finSupp .0. <-> w finSupp .0. ) )
22	21	elrab 3363	. . . . . . 7 |- ( w e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } <-> ( w e. ( ( Base ` R ) ^m NN0 ) /\ w finSupp .0. ) )
23	22	simprbi 480	. . . . . 6 |- ( w e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } -> w finSupp .0. )
24	20, 23	simplbiim 659	. . . . 5 |- ( w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) -> w finSupp .0. )
25	24	rgen 2922	. . . 4 |- A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) w finSupp .0.
26	25	a1i 11	. . 3 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) w finSupp .0. )
27		fsuppmapnn0fiub0 12793	. . . 4 |- ( ( ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) C_ ( ( Base ` R ) ^m NN0 ) /\ ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) e. Fin /\ .0. e. _V ) -> ( A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) w finSupp .0. -> E. s e. NN0 A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) ) )
28	27	imp 445	. . 3 |- ( ( ( ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) C_ ( ( Base ` R ) ^m NN0 ) /\ ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) e. Fin /\ .0. e. _V ) /\ A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) w finSupp .0. ) -> E. s e. NN0 A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) )
29	5, 15, 19, 26, 28	syl31anc 1329	. 2 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> E. s e. NN0 A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) )
30		opelxpi 5148	. . . . . . . . . . . . . . 15 |- ( ( i e. N /\ j e. N ) -> <. i , j >. e. ( N X. N ) )
31		df-ov 6653	. . . . . . . . . . . . . . . . . 18 |- ( i M j ) = ( M ` <. i , j >. )
32	31	fveq2i 6194	. . . . . . . . . . . . . . . . 17 |- (coe1 ` ( i M j ) ) = (coe1 ` ( M ` <. i , j >. ) )
33		fvex 6201	. . . . . . . . . . . . . . . . . 18 |- (coe1 ` ( M ` <. i , j >. ) ) e. _V
34	33	snid 4208	. . . . . . . . . . . . . . . . 17 |- (coe1 ` ( M ` <. i , j >. ) ) e. { (coe1 ` ( M ` <. i , j >. ) ) }
35	32, 34	eqeltri 2697	. . . . . . . . . . . . . . . 16 |- (coe1 ` ( i M j ) ) e. { (coe1 ` ( M ` <. i , j >. ) ) }
36	35	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( i e. N /\ j e. N ) -> (coe1 ` ( i M j ) ) e. { (coe1 ` ( M ` <. i , j >. ) ) } )
37		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( u = <. i , j >. -> ( M ` u ) = ( M ` <. i , j >. ) )
38	37	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( u = <. i , j >. -> (coe1 ` ( M ` u ) ) = (coe1 ` ( M ` <. i , j >. ) ) )
39	38	sneqd 4189	. . . . . . . . . . . . . . . 16 |- ( u = <. i , j >. -> { (coe1 ` ( M ` u ) ) } = { (coe1 ` ( M ` <. i , j >. ) ) } )
40	39	eliuni 4526	. . . . . . . . . . . . . . 15 |- ( ( <. i , j >. e. ( N X. N ) /\ (coe1 ` ( i M j ) ) e. { (coe1 ` ( M ` <. i , j >. ) ) } ) -> (coe1 ` ( i M j ) ) e. U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } )
41	30, 36, 40	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( i e. N /\ j e. N ) -> (coe1 ` ( i M j ) ) e. U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } )
42	41	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> (coe1 ` ( i M j ) ) e. U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } )
43		pmatcoe1fsupp.c	. . . . . . . . . . . . . . . 16 |- C = ( N Mat P )
44		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( Base ` P ) = ( Base ` P )
45		pmatcoe1fsupp.b	. . . . . . . . . . . . . . . 16 |- B = ( Base ` C )
46		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> i e. N )
47		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> j e. N )
48	45	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 |- ( M e. B <-> M e. ( Base ` C ) )
49	48	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( M e. B -> M e. ( Base ` C ) )
50	49	3ad2ant3 1084	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> M e. ( Base ` C ) )
51	50	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> M e. ( Base ` C ) )
52	51, 45	syl6eleqr 2712	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> M e. B )
53	43, 44, 45, 46, 47, 52	matecld 20232	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( i M j ) e. ( Base ` P ) )
54		eqid 2622	. . . . . . . . . . . . . . . 16 |- (coe1 ` ( i M j ) ) = (coe1 ` ( i M j ) )
55		pmatcoe1fsupp.p	. . . . . . . . . . . . . . . 16 |- P = (Poly1 ` R )
56		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( 0g ` R ) = ( 0g ` R )
57		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( Base ` R ) = ( Base ` R )
58	54, 44, 55, 56, 57	coe1fsupp 19584	. . . . . . . . . . . . . . 15 |- ( ( i M j ) e. ( Base ` P ) -> (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp ( 0g ` R ) } )
59	53, 58	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp ( 0g ` R ) } )
60	16	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> .0. = ( 0g ` R ) )
61	60	breq2d 4665	. . . . . . . . . . . . . . . . 17 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( v finSupp .0. <-> v finSupp ( 0g ` R ) ) )
62	61	rabbidv 3189	. . . . . . . . . . . . . . . 16 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } = { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp ( 0g ` R ) } )
63	62	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } <-> (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp ( 0g ` R ) } ) )
64	63	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } <-> (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp ( 0g ` R ) } ) )
65	59, 64	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> (coe1 ` ( i M j ) ) e. { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } )
66	42, 65	elind 3798	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> (coe1 ` ( i M j ) ) e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) )
67		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> x e. NN0 )
68		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( w = (coe1 ` ( i M j ) ) -> ( w ` z ) = ( (coe1 ` ( i M j ) ) ` z ) )
69	68	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( w = (coe1 ` ( i M j ) ) -> ( ( w ` z ) = .0. <-> ( (coe1 ` ( i M j ) ) ` z ) = .0. ) )
70	69	imbi2d 330	. . . . . . . . . . . . 13 |- ( w = (coe1 ` ( i M j ) ) -> ( ( s < z -> ( w ` z ) = .0. ) <-> ( s < z -> ( (coe1 ` ( i M j ) ) ` z ) = .0. ) ) )
71		breq2 4657	. . . . . . . . . . . . . 14 |- ( z = x -> ( s < z <-> s < x ) )
72		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( z = x -> ( (coe1 ` ( i M j ) ) ` z ) = ( (coe1 ` ( i M j ) ) ` x ) )
73	72	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( z = x -> ( ( (coe1 ` ( i M j ) ) ` z ) = .0. <-> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) )
74	71, 73	imbi12d 334	. . . . . . . . . . . . 13 |- ( z = x -> ( ( s < z -> ( (coe1 ` ( i M j ) ) ` z ) = .0. ) <-> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
75	70, 74	rspc2v 3322	. . . . . . . . . . . 12 |- ( ( (coe1 ` ( i M j ) ) e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) /\ x e. NN0 ) -> ( A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) -> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
76	66, 67, 75	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) -> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
77	76	ex 450	. . . . . . . . . 10 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) -> ( ( i e. N /\ j e. N ) -> ( A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) -> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) ) )
78	77	com23 86	. . . . . . . . 9 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ x e. NN0 ) -> ( A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) -> ( ( i e. N /\ j e. N ) -> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) ) )
79	78	impancom 456	. . . . . . . 8 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) ) -> ( x e. NN0 -> ( ( i e. N /\ j e. N ) -> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) ) )
80	79	imp 445	. . . . . . 7 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) ) /\ x e. NN0 ) -> ( ( i e. N /\ j e. N ) -> ( s < x -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
81	80	com23 86	. . . . . 6 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) ) /\ x e. NN0 ) -> ( s < x -> ( ( i e. N /\ j e. N ) -> ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
82	81	ralrimdvv 2973	. . . . 5 |- ( ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) ) /\ x e. NN0 ) -> ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = .0. ) )
83	82	ralrimiva 2966	. . . 4 |- ( ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) /\ A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) ) -> A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = .0. ) )
84	83	ex 450	. . 3 |- ( ( ( N e. Fin /\ R e. Ring /\ M e. B ) /\ s e. NN0 ) -> ( A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) -> A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
85	84	reximdva 3017	. 2 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( E. s e. NN0 A. w e. ( U_ u e. ( N X. N ) { (coe1 ` ( M ` u ) ) } i^i { v e. ( ( Base ` R ) ^m NN0 ) | v finSupp .0. } ) A. z e. NN0 ( s < z -> ( w ` z ) = .0. ) -> E. s e. NN0 A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = .0. ) ) )
86	29, 85	mpd 15	1 |- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> E. s e. NN0 A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   < clt 10074   NN0cn0 11292   Basecbs 15857   0gc0g 16100   Ringcrg 18547  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213

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-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-rmo 2920  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-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-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553  df-dsmm 20076  df-frlm 20091  df-mat 20214

This theorem is referenced by:  decpmataa0  20573  decpmatmulsumfsupp  20578  pmatcollpw2lem  20582  pm2mpmhmlem1  20623