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Mirrors > Home > MPE Home > Th. List > mat2pmatf | Structured version Visualization version GIF version |
Description: The matrix transformation is a function from the matrices to the polynomial matrices. (Contributed by AV, 27-Oct-2019.) |
Ref | Expression |
---|---|
mat2pmatbas.t | β’ π = (π matToPolyMat π ) |
mat2pmatbas.a | β’ π΄ = (π Mat π ) |
mat2pmatbas.b | β’ π΅ = (Baseβπ΄) |
mat2pmatbas.p | β’ π = (Poly1βπ ) |
mat2pmatbas.c | β’ πΆ = (π Mat π) |
mat2pmatbas0.h | β’ π» = (BaseβπΆ) |
Ref | Expression |
---|---|
mat2pmatf | β’ ((π β Fin β§ π β Ring) β π:π΅βΆπ») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . . 5 β’ ((π β Fin β§ π β Ring) β π β Fin) | |
2 | 1, 1 | jca 510 | . . . 4 β’ ((π β Fin β§ π β Ring) β (π β Fin β§ π β Fin)) |
3 | 2 | adantr 479 | . . 3 β’ (((π β Fin β§ π β Ring) β§ π β π΅) β (π β Fin β§ π β Fin)) |
4 | mpoexga 8080 | . . 3 β’ ((π β Fin β§ π β Fin) β (π₯ β π, π¦ β π β¦ ((algScβπ)β(π₯ππ¦))) β V) | |
5 | 3, 4 | syl 17 | . 2 β’ (((π β Fin β§ π β Ring) β§ π β π΅) β (π₯ β π, π¦ β π β¦ ((algScβπ)β(π₯ππ¦))) β V) |
6 | mat2pmatbas.t | . . 3 β’ π = (π matToPolyMat π ) | |
7 | mat2pmatbas.a | . . 3 β’ π΄ = (π Mat π ) | |
8 | mat2pmatbas.b | . . 3 β’ π΅ = (Baseβπ΄) | |
9 | mat2pmatbas.p | . . 3 β’ π = (Poly1βπ ) | |
10 | eqid 2725 | . . 3 β’ (algScβπ) = (algScβπ) | |
11 | 6, 7, 8, 9, 10 | mat2pmatfval 22643 | . 2 β’ ((π β Fin β§ π β Ring) β π = (π β π΅ β¦ (π₯ β π, π¦ β π β¦ ((algScβπ)β(π₯ππ¦))))) |
12 | mat2pmatbas.c | . . . 4 β’ πΆ = (π Mat π) | |
13 | mat2pmatbas0.h | . . . 4 β’ π» = (BaseβπΆ) | |
14 | 6, 7, 8, 9, 12, 13 | mat2pmatbas0 22647 | . . 3 β’ ((π β Fin β§ π β Ring β§ π β π΅) β (πβπ) β π») |
15 | 14 | 3expa 1115 | . 2 β’ (((π β Fin β§ π β Ring) β§ π β π΅) β (πβπ) β π») |
16 | 5, 11, 15 | fmpt2d 7129 | 1 β’ ((π β Fin β§ π β Ring) β π:π΅βΆπ») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 βΆwf 6539 βcfv 6543 (class class class)co 7416 β cmpo 7418 Fincfn 8962 Basecbs 17179 Ringcrg 20177 algSccascl 21790 Poly1cpl1 22104 Mat cmat 22325 matToPolyMat cmat2pmat 22624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-ascl 21793 df-psr 21846 df-mpl 21848 df-opsr 21850 df-psr1 22107 df-ply1 22109 df-mat 22326 df-mat2pmat 22627 |
This theorem is referenced by: mat2pmatf1 22649 mat2pmatghm 22650 mat2pmatmhm 22653 |
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