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Mirrors > Home > MPE Home > Th. List > pm2mprngiso | Structured version Visualization version GIF version |
Description: The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.) |
Ref | Expression |
---|---|
pm2mpmhm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pm2mpmhm.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pm2mpmhm.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pm2mpmhm.q | ⊢ 𝑄 = (Poly1‘𝐴) |
pm2mpmhm.t | ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) |
Ref | Expression |
---|---|
pm2mprngiso | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpmhm.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pm2mpmhm.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | pm2mpmhm.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | pm2mpmhm.q | . . 3 ⊢ 𝑄 = (Poly1‘𝐴) | |
5 | pm2mpmhm.t | . . 3 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
6 | 1, 2, 3, 4, 5 | pm2mprhm 21585 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄)) |
7 | eqid 2739 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | eqid 2739 | . . 3 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄) | |
9 | eqid 2739 | . . 3 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄)) | |
10 | eqid 2739 | . . 3 ⊢ (var1‘𝐴) = (var1‘𝐴) | |
11 | eqid 2739 | . . 3 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
12 | 1, 2, 7, 8, 9, 10, 3, 4, 11, 5 | pm2mpf1o 21579 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:(Base‘𝐶)–1-1-onto→(Base‘𝑄)) |
13 | 1, 2 | pmatring 21456 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
14 | 3 | matring 21207 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
15 | 4 | ply1ring 21036 | . . . 4 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring) |
16 | 14, 15 | syl 17 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring) |
17 | 7, 11 | isrim 19620 | . . 3 ⊢ ((𝐶 ∈ Ring ∧ 𝑄 ∈ Ring) → (𝑇 ∈ (𝐶 RingIso 𝑄) ↔ (𝑇 ∈ (𝐶 RingHom 𝑄) ∧ 𝑇:(Base‘𝐶)–1-1-onto→(Base‘𝑄)))) |
18 | 13, 16, 17 | syl2anc 587 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇 ∈ (𝐶 RingIso 𝑄) ↔ (𝑇 ∈ (𝐶 RingHom 𝑄) ∧ 𝑇:(Base‘𝐶)–1-1-onto→(Base‘𝑄)))) |
19 | 6, 12, 18 | mpbir2and 713 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7183 Fincfn 8568 Basecbs 16599 ·𝑠 cvsca 16685 .gcmg 18355 mulGrpcmgp 19371 Ringcrg 19429 RingHom crh 19599 RingIso crs 19600 var1cv1 20964 Poly1cpl1 20965 Mat cmat 21171 pMatToMatPoly cpm2mp 21556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-ot 4535 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 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 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-ofr 7439 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-sup 8992 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-fz 12995 df-fzo 13138 df-seq 13474 df-hash 13796 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-hom 16705 df-cco 16706 df-0g 16831 df-gsum 16832 df-prds 16837 df-pws 16839 df-mre 16973 df-mrc 16974 df-acs 16976 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-mulg 18356 df-subg 18407 df-ghm 18487 df-cntz 18578 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-srg 19388 df-ring 19431 df-rnghom 19602 df-rngiso 19603 df-subrg 19665 df-lmod 19768 df-lss 19836 df-sra 20076 df-rgmod 20077 df-dsmm 20561 df-frlm 20576 df-ascl 20684 df-psr 20735 df-mvr 20736 df-mpl 20737 df-opsr 20739 df-psr1 20968 df-vr1 20969 df-ply1 20970 df-coe1 20971 df-mamu 21150 df-mat 21172 df-decpmat 21527 df-pm2mp 21557 |
This theorem is referenced by: pmmpric 21587 |
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