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Mirrors > Home > MPE Home > Th. List > pmatcollpw | Structured version Visualization version Unicode version |
Description: Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.) |
Ref | Expression |
---|---|
pmatcollpw.p | Poly1 |
pmatcollpw.c | Mat |
pmatcollpw.b | |
pmatcollpw.m | |
pmatcollpw.e | .gmulGrp |
pmatcollpw.x | var1 |
pmatcollpw.t | matToPolyMat |
Ref | Expression |
---|---|
pmatcollpw | g decompPMat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 18558 | . . 3 | |
2 | pmatcollpw.p | . . . 4 Poly1 | |
3 | pmatcollpw.c | . . . 4 Mat | |
4 | pmatcollpw.b | . . . 4 | |
5 | eqid 2622 | . . . 4 | |
6 | pmatcollpw.e | . . . 4 .gmulGrp | |
7 | pmatcollpw.x | . . . 4 var1 | |
8 | 2, 3, 4, 5, 6, 7 | pmatcollpw2 20583 | . . 3 g decompPMat |
9 | 1, 8 | syl3an2 1360 | . 2 g decompPMat |
10 | eqidd 2623 | . . . . . . . 8 decompPMat decompPMat | |
11 | oveq12 6659 | . . . . . . . . . 10 decompPMat decompPMat | |
12 | 11 | oveq1d 6665 | . . . . . . . . 9 decompPMat decompPMat |
13 | 12 | adantl 482 | . . . . . . . 8 decompPMat decompPMat |
14 | simprl 794 | . . . . . . . 8 | |
15 | simpr 477 | . . . . . . . . 9 | |
16 | 15 | adantl 482 | . . . . . . . 8 |
17 | simp2 1062 | . . . . . . . . . . . 12 | |
18 | 17 | adantr 481 | . . . . . . . . . . 11 |
19 | 18, 1 | syl 17 | . . . . . . . . . 10 |
20 | 19 | adantr 481 | . . . . . . . . 9 |
21 | eqid 2622 | . . . . . . . . . 10 Mat Mat | |
22 | eqid 2622 | . . . . . . . . . 10 | |
23 | eqid 2622 | . . . . . . . . . 10 Mat Mat | |
24 | simp3 1063 | . . . . . . . . . . . . 13 | |
25 | 24 | adantr 481 | . . . . . . . . . . . 12 |
26 | simpr 477 | . . . . . . . . . . . 12 | |
27 | 2, 3, 4, 21, 23 | decpmatcl 20572 | . . . . . . . . . . . 12 decompPMat Mat |
28 | 18, 25, 26, 27 | syl3anc 1326 | . . . . . . . . . . 11 decompPMat Mat |
29 | 28 | adantr 481 | . . . . . . . . . 10 decompPMat Mat |
30 | 21, 22, 23, 14, 16, 29 | matecld 20232 | . . . . . . . . 9 decompPMat |
31 | simplr 792 | . . . . . . . . 9 | |
32 | eqid 2622 | . . . . . . . . . 10 mulGrp mulGrp | |
33 | eqid 2622 | . . . . . . . . . 10 | |
34 | 22, 2, 7, 5, 32, 6, 33 | ply1tmcl 19642 | . . . . . . . . 9 decompPMat decompPMat |
35 | 20, 30, 31, 34 | syl3anc 1326 | . . . . . . . 8 decompPMat |
36 | 10, 13, 14, 16, 35 | ovmpt2d 6788 | . . . . . . 7 decompPMat decompPMat |
37 | pmatcollpw.m | . . . . . . . . 9 | |
38 | pmatcollpw.t | . . . . . . . . 9 matToPolyMat | |
39 | 2, 3, 4, 37, 6, 7, 38 | pmatcollpwlem 20585 | . . . . . . . 8 decompPMat decompPMat |
40 | 39 | 3expb 1266 | . . . . . . 7 decompPMat decompPMat |
41 | 36, 40 | eqtrd 2656 | . . . . . 6 decompPMat decompPMat |
42 | 41 | ralrimivva 2971 | . . . . 5 decompPMat decompPMat |
43 | simpl1 1064 | . . . . . . 7 | |
44 | 2 | ply1ring 19618 | . . . . . . . . . 10 |
45 | 1, 44 | syl 17 | . . . . . . . . 9 |
46 | 45 | 3ad2ant2 1083 | . . . . . . . 8 |
47 | 46 | adantr 481 | . . . . . . 7 |
48 | 19 | 3ad2ant1 1082 | . . . . . . . 8 |
49 | simp2 1062 | . . . . . . . . 9 | |
50 | simp3 1063 | . . . . . . . . 9 | |
51 | 28 | 3ad2ant1 1082 | . . . . . . . . 9 decompPMat Mat |
52 | 21, 22, 23, 49, 50, 51 | matecld 20232 | . . . . . . . 8 decompPMat |
53 | 26 | 3ad2ant1 1082 | . . . . . . . 8 |
54 | 22, 2, 7, 5, 32, 6, 33 | ply1tmcl 19642 | . . . . . . . 8 decompPMat decompPMat |
55 | 48, 52, 53, 54 | syl3anc 1326 | . . . . . . 7 decompPMat |
56 | 3, 33, 4, 43, 47, 55 | matbas2d 20229 | . . . . . 6 decompPMat |
57 | 1 | 3ad2ant2 1083 | . . . . . . . 8 |
58 | 2, 7, 32, 6, 33 | ply1moncl 19641 | . . . . . . . 8 |
59 | 57, 58 | sylan 488 | . . . . . . 7 |
60 | 57 | adantr 481 | . . . . . . . . 9 |
61 | 38, 21, 23, 2, 3 | mat2pmatbas 20531 | . . . . . . . . 9 decompPMat Mat decompPMat |
62 | 43, 60, 28, 61 | syl3anc 1326 | . . . . . . . 8 decompPMat |
63 | 62, 4 | syl6eleqr 2712 | . . . . . . 7 decompPMat |
64 | 33, 3, 4, 37 | matvscl 20237 | . . . . . . 7 decompPMat decompPMat |
65 | 43, 47, 59, 63, 64 | syl22anc 1327 | . . . . . 6 decompPMat |
66 | 3, 4 | eqmat 20230 | . . . . . 6 decompPMat decompPMat decompPMat decompPMat decompPMat decompPMat |
67 | 56, 65, 66 | syl2anc 693 | . . . . 5 decompPMat decompPMat decompPMat decompPMat |
68 | 42, 67 | mpbird 247 | . . . 4 decompPMat decompPMat |
69 | 68 | mpteq2dva 4744 | . . 3 decompPMat decompPMat |
70 | 69 | oveq2d 6666 | . 2 g decompPMat g decompPMat |
71 | 9, 70 | eqtrd 2656 | 1 g decompPMat |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cmpt 4729 cfv 5888 (class class class)co 6650 cmpt2 6652 cfn 7955 cn0 11292 cbs 15857 cvsca 15945 g cgsu 16101 .gcmg 17540 mulGrpcmgp 18489 crg 18547 ccrg 18548 var1cv1 19546 Poly1cpl1 19547 Mat cmat 20213 matToPolyMat cmat2pmat 20509 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-fal 1489 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-ofr 6898 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-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 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-srg 18506 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-assa 19312 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-dsmm 20076 df-frlm 20091 df-mat 20214 df-mat2pmat 20512 df-decpmat 20568 |
This theorem is referenced by: pmatcollpwfi 20587 pmatcollpw3 20589 pmatcollpwscmat 20596 |
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