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Mirrors > Home > MPE Home > Th. List > pmmpric | Structured version Visualization version GIF version |
Description: The ring of polynomial matrices over a ring is isomorphic to the ring of polynomials over matrices of the same dimension over the same ring. (Contributed by AV, 30-Dec-2019.) |
Ref | Expression |
---|---|
pmmpric.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pmmpric.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pmmpric.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pmmpric.q | ⊢ 𝑄 = (Poly1‘𝐴) |
Ref | Expression |
---|---|
pmmpric | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ≃𝑟 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmmpric.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pmmpric.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | pmmpric.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | pmmpric.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
5 | eqid 2736 | . . . 4 ⊢ (𝑁 pMatToMatPoly 𝑅) = (𝑁 pMatToMatPoly 𝑅) | |
6 | 1, 2, 3, 4, 5 | pm2mprngiso 22016 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 pMatToMatPoly 𝑅) ∈ (𝐶 RingIso 𝑄)) |
7 | 6 | ne0d 4275 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐶 RingIso 𝑄) ≠ ∅) |
8 | brric 20033 | . 2 ⊢ (𝐶 ≃𝑟 𝑄 ↔ (𝐶 RingIso 𝑄) ≠ ∅) | |
9 | 7, 8 | sylibr 233 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ≃𝑟 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 ∅c0 4262 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 Fincfn 8764 Ringcrg 19828 RingIso crs 20002 ≃𝑟 cric 20003 Poly1cpl1 21393 Mat cmat 21599 pMatToMatPoly cpm2mp 21986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-ot 4574 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-ofr 7566 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-pm 8649 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-sup 9245 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-fz 13286 df-fzo 13429 df-seq 13768 df-hash 14091 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-hom 17031 df-cco 17032 df-0g 17197 df-gsum 17198 df-prds 17203 df-pws 17205 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-mhm 18475 df-submnd 18476 df-grp 18625 df-minusg 18626 df-sbg 18627 df-mulg 18746 df-subg 18797 df-ghm 18877 df-cntz 18968 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-srg 19787 df-ring 19830 df-rnghom 20004 df-rngiso 20005 df-ric 20007 df-subrg 20067 df-lmod 20170 df-lss 20239 df-sra 20479 df-rgmod 20480 df-dsmm 20984 df-frlm 20999 df-ascl 21107 df-psr 21157 df-mvr 21158 df-mpl 21159 df-opsr 21161 df-psr1 21396 df-vr1 21397 df-ply1 21398 df-coe1 21399 df-mamu 21578 df-mat 21600 df-decpmat 21957 df-pm2mp 21987 |
This theorem is referenced by: (None) |
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