Theorem decpmataa0 20573

Description: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power is 0 for almost all powers. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 3-Dec-2019.)

Hypotheses

Ref	Expression
decpmate.p	|- P = (Poly1 ` R )
decpmate.c	|- C = ( N Mat P )
decpmate.b	|- B = ( Base ` C )
decpmatcl.a	|- A = ( N Mat R )
decpmatfsupp.0	|- .0. = ( 0g ` A )

Assertion

Ref	Expression
decpmataa0	|- ( ( R e. Ring /\ M e. B ) -> E. s e. NN0 A. x e. NN0 ( s < x -> ( M decompPMat x ) = .0. ) )

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

Allowed substitution hints:   A( x, s)   C( x, s)   P( x, s)

Proof of Theorem decpmataa0

Dummy variables i j a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		decpmate.c	. . . . . 6 |- C = ( N Mat P )
2		decpmate.b	. . . . . 6 |- B = ( Base ` C )
3	1, 2	matrcl 20218	. . . . 5 |- ( M e. B -> ( N e. Fin /\ P e. _V ) )
4	3	simpld 475	. . . 4 |- ( M e. B -> N e. Fin )
5	4	adantl 482	. . 3 |- ( ( R e. Ring /\ M e. B ) -> N e. Fin )
6		simpl 473	. . 3 |- ( ( R e. Ring /\ M e. B ) -> R e. Ring )
7		simpr 477	. . 3 |- ( ( R e. Ring /\ M e. B ) -> M e. B )
8		decpmate.p	. . . 4 |- P = (Poly1 ` R )
9		eqid 2622	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
10	8, 1, 2, 9	pmatcoe1fsupp 20506	. . 3 |- ( ( 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 ) = ( 0g ` R ) ) )
11	5, 6, 7, 10	syl3anc 1326	. 2 |- ( ( 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 ) = ( 0g ` R ) ) )
12		decpmatcl.a	. . . . . . . . 9 |- A = ( N Mat R )
13		eqid 2622	. . . . . . . . 9 |- ( Base ` A ) = ( Base ` A )
14	8, 1, 2, 12, 13	decpmatcl 20572	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B /\ x e. NN0 ) -> ( M decompPMat x ) e. ( Base ` A ) )
15	14	3expa 1265	. . . . . . 7 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> ( M decompPMat x ) e. ( Base ` A ) )
16	5, 6	jca 554	. . . . . . . . 9 |- ( ( R e. Ring /\ M e. B ) -> ( N e. Fin /\ R e. Ring ) )
17	12	matring 20249	. . . . . . . . 9 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
18		decpmatfsupp.0	. . . . . . . . . 10 |- .0. = ( 0g ` A )
19	13, 18	ring0cl 18569	. . . . . . . . 9 |- ( A e. Ring -> .0. e. ( Base ` A ) )
20	16, 17, 19	3syl 18	. . . . . . . 8 |- ( ( R e. Ring /\ M e. B ) -> .0. e. ( Base ` A ) )
21	20	adantr 481	. . . . . . 7 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> .0. e. ( Base ` A ) )
22	12, 13	eqmat 20230	. . . . . . 7 |- ( ( ( M decompPMat x ) e. ( Base ` A ) /\ .0. e. ( Base ` A ) ) -> ( ( M decompPMat x ) = .0. <-> A. i e. N A. j e. N ( i ( M decompPMat x ) j ) = ( i .0. j ) ) )
23	15, 21, 22	syl2anc 693	. . . . . 6 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> ( ( M decompPMat x ) = .0. <-> A. i e. N A. j e. N ( i ( M decompPMat x ) j ) = ( i .0. j ) ) )
24		df-3an 1039	. . . . . . . . 9 |- ( ( R e. Ring /\ M e. B /\ x e. NN0 ) <-> ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) )
25	8, 1, 2	decpmate 20571	. . . . . . . . 9 |- ( ( ( R e. Ring /\ M e. B /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( i ( M decompPMat x ) j ) = ( (coe1 ` ( i M j ) ) ` x ) )
26	24, 25	sylanbr 490	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( i ( M decompPMat x ) j ) = ( (coe1 ` ( i M j ) ) ` x ) )
27	16	adantr 481	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> ( N e. Fin /\ R e. Ring ) )
28	27	adantr 481	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( N e. Fin /\ R e. Ring ) )
29	12, 9	mat0op 20225	. . . . . . . . . . 11 |- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` A ) = ( a e. N , b e. N |-> ( 0g ` R ) ) )
30	18, 29	syl5eq 2668	. . . . . . . . . 10 |- ( ( N e. Fin /\ R e. Ring ) -> .0. = ( a e. N , b e. N |-> ( 0g ` R ) ) )
31	28, 30	syl 17	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> .0. = ( a e. N , b e. N |-> ( 0g ` R ) ) )
32		eqidd 2623	. . . . . . . . 9 |- ( ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) /\ ( a = i /\ b = j ) ) -> ( 0g ` R ) = ( 0g ` R ) )
33		simpl 473	. . . . . . . . . 10 |- ( ( i e. N /\ j e. N ) -> i e. N )
34	33	adantl 482	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> i e. N )
35		simpr 477	. . . . . . . . . 10 |- ( ( i e. N /\ j e. N ) -> j e. N )
36	35	adantl 482	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> j e. N )
37		fvexd 6203	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( 0g ` R ) e. _V )
38	31, 32, 34, 36, 37	ovmpt2d 6788	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( i .0. j ) = ( 0g ` R ) )
39	26, 38	eqeq12d 2637	. . . . . . 7 |- ( ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) /\ ( i e. N /\ j e. N ) ) -> ( ( i ( M decompPMat x ) j ) = ( i .0. j ) <-> ( (coe1 ` ( i M j ) ) ` x ) = ( 0g ` R ) ) )
40	39	2ralbidva 2988	. . . . . 6 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> ( A. i e. N A. j e. N ( i ( M decompPMat x ) j ) = ( i .0. j ) <-> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = ( 0g ` R ) ) )
41	23, 40	bitrd 268	. . . . 5 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> ( ( M decompPMat x ) = .0. <-> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = ( 0g ` R ) ) )
42	41	imbi2d 330	. . . 4 |- ( ( ( R e. Ring /\ M e. B ) /\ x e. NN0 ) -> ( ( s < x -> ( M decompPMat x ) = .0. ) <-> ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = ( 0g ` R ) ) ) )
43	42	ralbidva 2985	. . 3 |- ( ( R e. Ring /\ M e. B ) -> ( A. x e. NN0 ( s < x -> ( M decompPMat x ) = .0. ) <-> A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = ( 0g ` R ) ) ) )
44	43	rexbidv 3052	. 2 |- ( ( R e. Ring /\ M e. B ) -> ( E. s e. NN0 A. x e. NN0 ( s < x -> ( M decompPMat x ) = .0. ) <-> E. s e. NN0 A. x e. NN0 ( s < x -> A. i e. N A. j e. N ( (coe1 ` ( i M j ) ) ` x ) = ( 0g ` R ) ) ) )
45	11, 44	mpbird 247	1 |- ( ( R e. Ring /\ M e. B ) -> E. s e. NN0 A. x e. NN0 ( s < x -> ( M decompPMat x ) = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   < clt 10074   NN0cn0 11292   Basecbs 15857   0gc0g 16100   Ringcrg 18547  Poly₁cpl1 19547  coe₁cco1 19548   Mat cmat 20213   decompPMat cdecpmat 20567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-iin 4523  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-se 5074  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-isom 5897  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-oi 8415  df-card 8765  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-fzo 12466  df-seq 12802  df-hash 13118  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-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  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-mamu 20190  df-mat 20214  df-decpmat 20568

This theorem is referenced by:  decpmatfsupp  20574  pmatcollpwfi  20587