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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 2821 | . . . 4 ⊢ (𝑁 pMatToMatPoly 𝑅) = (𝑁 pMatToMatPoly 𝑅) | |
6 | 1, 2, 3, 4, 5 | pm2mprngiso 21424 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 pMatToMatPoly 𝑅) ∈ (𝐶 RingIso 𝑄)) |
7 | 6 | ne0d 4301 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐶 RingIso 𝑄) ≠ ∅) |
8 | brric 19493 | . 2 ⊢ (𝐶 ≃𝑟 𝑄 ↔ (𝐶 RingIso 𝑄) ≠ ∅) | |
9 | 7, 8 | sylibr 236 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ≃𝑟 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4291 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 Ringcrg 19291 RingIso crs 19459 ≃𝑟 cric 19460 Poly1cpl1 20339 Mat cmat 21010 pMatToMatPoly cpm2mp 21394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-srg 19250 df-ring 19293 df-rnghom 19461 df-rngiso 19462 df-ric 19464 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-vr1 20343 df-ply1 20344 df-coe1 20345 df-dsmm 20870 df-frlm 20885 df-mamu 20989 df-mat 21011 df-decpmat 21365 df-pm2mp 21395 |
This theorem is referenced by: (None) |
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