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Mirrors > Home > MPE Home > Th. List > pm2mpfval | Structured version Visualization version Unicode version |
Description: A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
Ref | Expression |
---|---|
pm2mpval.p | Poly1 |
pm2mpval.c | Mat |
pm2mpval.b | |
pm2mpval.m | |
pm2mpval.e | .gmulGrp |
pm2mpval.x | var1 |
pm2mpval.a | Mat |
pm2mpval.q | Poly1 |
pm2mpval.t | pMatToMatPoly |
Ref | Expression |
---|---|
pm2mpfval | g decompPMat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpval.p | . . . 4 Poly1 | |
2 | pm2mpval.c | . . . 4 Mat | |
3 | pm2mpval.b | . . . 4 | |
4 | pm2mpval.m | . . . 4 | |
5 | pm2mpval.e | . . . 4 .gmulGrp | |
6 | pm2mpval.x | . . . 4 var1 | |
7 | pm2mpval.a | . . . 4 Mat | |
8 | pm2mpval.q | . . . 4 Poly1 | |
9 | pm2mpval.t | . . . 4 pMatToMatPoly | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | pm2mpval 20600 | . . 3 g decompPMat |
11 | 10 | 3adant3 1081 | . 2 g decompPMat |
12 | oveq1 6657 | . . . . . 6 decompPMat decompPMat | |
13 | 12 | oveq1d 6665 | . . . . 5 decompPMat decompPMat |
14 | 13 | mpteq2dv 4745 | . . . 4 decompPMat decompPMat |
15 | 14 | oveq2d 6666 | . . 3 g decompPMat g decompPMat |
16 | 15 | adantl 482 | . 2 g decompPMat g decompPMat |
17 | simp3 1063 | . 2 | |
18 | ovexd 6680 | . 2 g decompPMat | |
19 | 11, 16, 17, 18 | fvmptd 6288 | 1 g decompPMat |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wceq 1483 wcel 1990 cvv 3200 cmpt 4729 cfv 5888 (class class class)co 6650 cfn 7955 cn0 11292 cbs 15857 cvsca 15945 g cgsu 16101 .gcmg 17540 mulGrpcmgp 18489 var1cv1 19546 Poly1cpl1 19547 Mat cmat 20213 decompPMat cdecpmat 20567 pMatToMatPoly cpm2mp 20597 |
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-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-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-reu 2919 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm2mp 20598 |
This theorem is referenced by: pm2mpcl 20602 pm2mpf1 20604 pm2mpcoe1 20605 idpm2idmp 20606 mp2pm2mp 20616 pm2mpghm 20621 pm2mpmhmlem2 20624 monmat2matmon 20629 |
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