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Mirrors > Home > MPE Home > Th. List > m2cpminvid2 | Structured version Visualization version Unicode version |
Description: The transformation applied to the inverse transformation of a constant polynomial matrix over the ring results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.) |
Ref | Expression |
---|---|
m2cpminvid2.s | ConstPolyMat |
m2cpminvid2.i | cPolyMatToMat |
m2cpminvid2.t | matToPolyMat |
Ref | Expression |
---|---|
m2cpminvid2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m2cpminvid2.i | . . . 4 cPolyMatToMat | |
2 | m2cpminvid2.s | . . . 4 ConstPolyMat | |
3 | 1, 2 | cpm2mval 20555 | . . 3 coe1 |
4 | 3 | fveq2d 6195 | . 2 coe1 |
5 | simp1 1061 | . . . 4 | |
6 | simp2 1062 | . . . 4 | |
7 | eqid 2622 | . . . . 5 Mat Mat | |
8 | eqid 2622 | . . . . 5 | |
9 | eqid 2622 | . . . . 5 Mat Mat | |
10 | eqid 2622 | . . . . . . 7 Mat Poly1 Mat Poly1 | |
11 | eqid 2622 | . . . . . . 7 Poly1 Poly1 | |
12 | eqid 2622 | . . . . . . 7 Mat Poly1 Mat Poly1 | |
13 | simp2 1062 | . . . . . . 7 | |
14 | simp3 1063 | . . . . . . 7 | |
15 | eqid 2622 | . . . . . . . . 9 Poly1 Poly1 | |
16 | 2, 15, 10, 12 | cpmatpmat 20515 | . . . . . . . 8 Mat Poly1 |
17 | 16 | 3ad2ant1 1082 | . . . . . . 7 Mat Poly1 |
18 | 10, 11, 12, 13, 14, 17 | matecld 20232 | . . . . . 6 Poly1 |
19 | 0nn0 11307 | . . . . . 6 | |
20 | eqid 2622 | . . . . . . 7 coe1 coe1 | |
21 | 20, 11, 15, 8 | coe1fvalcl 19582 | . . . . . 6 Poly1 coe1 |
22 | 18, 19, 21 | sylancl 694 | . . . . 5 coe1 |
23 | 7, 8, 9, 5, 6, 22 | matbas2d 20229 | . . . 4 coe1 Mat |
24 | m2cpminvid2.t | . . . . 5 matToPolyMat | |
25 | eqid 2622 | . . . . 5 algScPoly1 algScPoly1 | |
26 | 24, 7, 9, 15, 25 | mat2pmatval 20529 | . . . 4 coe1 Mat coe1 algScPoly1 coe1 |
27 | 5, 6, 23, 26 | syl3anc 1326 | . . 3 coe1 algScPoly1 coe1 |
28 | eqidd 2623 | . . . . . 6 coe1 coe1 | |
29 | oveq12 6659 | . . . . . . . . 9 | |
30 | 29 | fveq2d 6195 | . . . . . . . 8 coe1 coe1 |
31 | 30 | fveq1d 6193 | . . . . . . 7 coe1 coe1 |
32 | 31 | adantl 482 | . . . . . 6 coe1 coe1 |
33 | simp2 1062 | . . . . . 6 | |
34 | simp3 1063 | . . . . . 6 | |
35 | fvexd 6203 | . . . . . 6 coe1 | |
36 | 28, 32, 33, 34, 35 | ovmpt2d 6788 | . . . . 5 coe1 coe1 |
37 | 36 | fveq2d 6195 | . . . 4 algScPoly1 coe1 algScPoly1coe1 |
38 | 37 | mpt2eq3dva 6719 | . . 3 algScPoly1 coe1 algScPoly1coe1 |
39 | 27, 38 | eqtrd 2656 | . 2 coe1 algScPoly1coe1 |
40 | 2, 15 | m2cpminvid2lem 20559 | . . . . . 6 coe1algScPoly1coe1 coe1 |
41 | simpl2 1065 | . . . . . . 7 | |
42 | simprl 794 | . . . . . . . . . 10 | |
43 | simprr 796 | . . . . . . . . . 10 | |
44 | 16 | adantr 481 | . . . . . . . . . 10 Mat Poly1 |
45 | 10, 11, 12, 42, 43, 44 | matecld 20232 | . . . . . . . . 9 Poly1 |
46 | 45, 19, 21 | sylancl 694 | . . . . . . . 8 coe1 |
47 | 15, 25, 8, 11 | ply1sclcl 19656 | . . . . . . . 8 coe1 algScPoly1coe1 Poly1 |
48 | 41, 46, 47 | syl2anc 693 | . . . . . . 7 algScPoly1coe1 Poly1 |
49 | eqid 2622 | . . . . . . . . 9 coe1algScPoly1coe1 coe1algScPoly1coe1 | |
50 | 15, 11, 49, 20 | ply1coe1eq 19668 | . . . . . . . 8 algScPoly1coe1 Poly1 Poly1 coe1algScPoly1coe1 coe1 algScPoly1coe1 |
51 | 50 | bicomd 213 | . . . . . . 7 algScPoly1coe1 Poly1 Poly1 algScPoly1coe1 coe1algScPoly1coe1 coe1 |
52 | 41, 48, 45, 51 | syl3anc 1326 | . . . . . 6 algScPoly1coe1 coe1algScPoly1coe1 coe1 |
53 | 40, 52 | mpbird 247 | . . . . 5 algScPoly1coe1 |
54 | 53 | ralrimivva 2971 | . . . 4 algScPoly1coe1 |
55 | eqidd 2623 | . . . . . . . 8 algScPoly1coe1 algScPoly1coe1 | |
56 | oveq12 6659 | . . . . . . . . . . . 12 | |
57 | 56 | fveq2d 6195 | . . . . . . . . . . 11 coe1 coe1 |
58 | 57 | fveq1d 6193 | . . . . . . . . . 10 coe1 coe1 |
59 | 58 | fveq2d 6195 | . . . . . . . . 9 algScPoly1coe1 algScPoly1coe1 |
60 | 59 | adantl 482 | . . . . . . . 8 algScPoly1coe1 algScPoly1coe1 |
61 | simplr 792 | . . . . . . . 8 | |
62 | simpr 477 | . . . . . . . 8 | |
63 | fvexd 6203 | . . . . . . . 8 algScPoly1coe1 | |
64 | 55, 60, 61, 62, 63 | ovmpt2d 6788 | . . . . . . 7 algScPoly1coe1 algScPoly1coe1 |
65 | 64 | eqeq1d 2624 | . . . . . 6 algScPoly1coe1 algScPoly1coe1 |
66 | 65 | anasss 679 | . . . . 5 algScPoly1coe1 algScPoly1coe1 |
67 | 66 | 2ralbidva 2988 | . . . 4 algScPoly1coe1 algScPoly1coe1 |
68 | 54, 67 | mpbird 247 | . . 3 algScPoly1coe1 |
69 | fvexd 6203 | . . . . 5 Poly1 | |
70 | 6 | 3ad2ant1 1082 | . . . . . 6 |
71 | 16 | 3ad2ant1 1082 | . . . . . . . 8 Mat Poly1 |
72 | 10, 11, 12, 33, 34, 71 | matecld 20232 | . . . . . . 7 Poly1 |
73 | eqid 2622 | . . . . . . . 8 coe1 coe1 | |
74 | 73, 11, 15, 8 | coe1fvalcl 19582 | . . . . . . 7 Poly1 coe1 |
75 | 72, 19, 74 | sylancl 694 | . . . . . 6 coe1 |
76 | 15, 25, 8, 11 | ply1sclcl 19656 | . . . . . 6 coe1 algScPoly1coe1 Poly1 |
77 | 70, 75, 76 | syl2anc 693 | . . . . 5 algScPoly1coe1 Poly1 |
78 | 10, 11, 12, 5, 69, 77 | matbas2d 20229 | . . . 4 algScPoly1coe1 Mat Poly1 |
79 | 10, 12 | eqmat 20230 | . . . 4 algScPoly1coe1 Mat Poly1 Mat Poly1 algScPoly1coe1 algScPoly1coe1 |
80 | 78, 16, 79 | syl2anc 693 | . . 3 algScPoly1coe1 algScPoly1coe1 |
81 | 68, 80 | mpbird 247 | . 2 algScPoly1coe1 |
82 | 4, 39, 81 | 3eqtrd 2660 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cvv 3200 cfv 5888 (class class class)co 6650 cmpt2 6652 cfn 7955 cc0 9936 cn0 11292 cbs 15857 crg 18547 algSccascl 19311 Poly1cpl1 19547 coe1cco1 19548 Mat cmat 20213 ConstPolyMat ccpmat 20508 matToPolyMat cmat2pmat 20509 cPolyMatToMat ccpmat2mat 20510 |
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-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 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-cpmat 20511 df-mat2pmat 20512 df-cpmat2mat 20513 |
This theorem is referenced by: m2cpmfo 20561 m2cpminv 20565 cpmadumatpoly 20688 cayhamlem4 20693 |
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